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OXFORD LOGIC GUIDES
General Editor: Dana Scott
THE
BOOLE-DE MORGAN
CORRESPONDENCE
1842-1864
G. C SMITH
Monash University
CLARENDON PRESS * OXFORD
1982
Oxford University Press, Walton Street, Oxford 0X2 6DP
London Glasgow New York Toronto
Delhi Bombay Calcutta Madras Karachi
Kaula Lumpur Singapore Hong Kong Tokyo
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and associate companies in
Beirut Berlin Ibadan Mexico City
Commentary ©G. C. Smith/f982
ACCESSION No,
^S\\ 3 Boo
Published in the United Slates "By
Oxford University Press, New York
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without
the prior permission of Oxford University Press
British Library Cataloguing in Publication Data
Boole, George
The Boole -De Morgan correspondence
1 842-1 864. ™ (Oxford logic guides)
1 , Boole, George 2. De Morgan, Augustus
3. Logic, Symbolic and mathematical
L Title II. De Morgan, Augustus
51L3.'092'2 B1574.B/
mjruQ-fittnKU
f
■- "■ * I *"t ^i,.
CLASS No.
'3HJ5
5
N
CATEGORY
K\O^M<\*
'3|3
Printed in Great Britain at the Alder) Press
Oxford London arid Northampton
Nothing gives so just an idea of an age
as genuine letters; nay history waits for
its last seal from them.
Horace Walpole to
Sir David Dalrymple,
30 November 1 76 1
ACKNOWLEDGEMENTS
I gratefully acknowledge permission to print the text of letters and other
manuscripts granted by the Director of the University of London Library and
the Librarians of the Royal Society and of University College London.
The first section of the bibliography includes detailed citation of the particular
manuscripts in these libraries that I have quoted. I also wish to express my
thanks to the staff of these libraries for facilitating the study and copying of the
documents.
Many persons have helped in various ways with information and advice and to
them I offer collective thanks. A few have given me a great deal of assistance and
these I thank individually: Ivor Grattan-Guinness, John Crossley, and
Anne-Marie Vandenberg.
G. C. S.
CONTENTS
Introduction 1
1. Getting acquainted: 1842-1845 7
2. Mathematical logic and Ireland: 1847-1850 17
3. Probabilities and eccentricity: 1851 40
4. The laws of thought and marriage: 1852-1856 56
5. Books old and new; and homoeopathy: 1859-1861 74
6. The controversy with Hamilton's successors; and the Jews: 1861—1862 85
7. From differential equations to spiritualism: 1863— 1864 106
8. Epilogue 118
Appendix: Boole's theorem on definite integration 124
Biographical notes 131
Bibliography: Manuscripts 136
Printed works 137
Index 153
INTRODUCTION
George Boole and Augustus De Morgan carried on a correspondence over a
period of some 22 years. About 90 of the letters between them have survived. 1
The major interest in the correspondence must be the exchange of ideas on
logical matters, because Boole and, to a lesser extent, De Morgan were innovators
in this field. But the letters also show their interest in other mathematical
matters: differential equations and probability being referred to not infrequently.
The letters also contain comments on a wide range of social, literary, political,
and religious matters. Nor are they devoid of purely personal interest: one might
instance Boole's initial unhappiness in Cork, his request to De Morgan to let him
know if there was any suitable post in England, and De Morgan's letter following
his recovery from pleurisy. Although in the main their correspondence was
conducted for serious reasons — to make requests for information or to answer
such requests — some letters are lighter in tone. De Morgan has an incisive style
and, when appropriate, shows a pleasing sense of humour: while Boole's rather
solemn style in his letters of the earlier years becomes increasingly more relaxed
as the years go by. Of the ninety letters Boole wrote sixty-six. However, while
Boole's letters are with few exceptions quite short, a few of De Morgan's extend
over 8 or more sides of notepaper. In consequence the balance is not quite so
heavily in Boole's favour as the numbers suggest. Nevertheless it seems to me
that the correspondence gives a more balanced and comprehensive view of Boole
and his work and personality than it does of De Morgan and his.
It would not be appropriate to include extended biographies of Boole and
De Morgan here. It is unfortunately the case that there has never been a full-scale
biography of Boole, and the biography of De Morgan written by his wife is
disappointing. 2 However, it may be helpful to record the salient facts of the lives
of Boole and De Morgan and to give some indication of their work, as this will
help the reader to understand the correspondence.
George Boole
Boole was born in humble circumstances in Lincoln on 2 November 1815. His
father, John Boole, was a cobbler who had a studious nature. John Boole's
interests lay in mathematics and astronomy; he constructed optical instruments;
he was also active in the Mechanics' Institute. George was the eldest child of the
2 INTRODUCTION
family. There was a sister, Mary Ann, born in 1818, and brothers William, born
in 1819, and Charles, born in 1821.
George Boole attended a primary school of the National Society and, later,
learned commercial subjects at, presumably, a private school of some kind. In
addition he extended his education by taking lessons in Latin from a bookseller
who was a friend of the family. Subsequently he taught himself Greek, French,
German, and Italian.
At the age of sixteen Boole became an usher (i.e. assistant master) at a school
in Doncaster — a town about 60 km from Lincoln. Within two years he returned
to work as a teacher either in Lincoln or in the nearby village of Waddington
until he moved to Cork in 1849. During this period Boole continued his study of
languages as well as commencing the study of theology with the intention of
entering the Church. He seems to have begun serious study of mathematics fairly
soon after beginning to teach; in 1832 he was reading Lacroix's Calcul
Differentiel. Later he read and learned from Poisson's Traite de Mecanique and
Lagrange's Calcul des Fonctions.
His reading, and investigations based upon his reading, led to his writing
papers which were published in the Cambridge Mathematical J T oumal from 1841 .
Thereby he became known to mathematicians in Cambridge and to a wider circle
following the publication of a large-scale paper on operational methods in the
Philosophical Transactions of the Royal Society in 1844. For this work he
received a Royal Medal of the Royal Society. In 1847 his seminal work, The
Mathematical Analysis of Logic, appeared. In 1849 he was appointed Professor
of Mathematics at Queen's College, Cork, where he remained until his death in
1864. While there, Boole met Mary Everest (1832-1916) whom he married in
1856; they had five daughters.
Topics discussed in Boole's early papers included linear transformations
(Boole was one of the originators of the theory of invariants); differential
equations, generally solved by an operational method; the evaluation of definite
and multiple integrals; and the calculus of variations.
After the publication of The Mathematical Analysis of Logic in 1847, Boole's
papers show an increasing interest in probability. Papers on probability pre-
dominate in the years 1851-7, although Boole continued to publish on the
topics mentioned in the previous paragraph. After the major paper in 1857
relating to the application of probabilities to 'the question of the combination of
testimonies or judgements' (Boole 1857ft) he published only two papers on
probabilities.
The years 1859 and 1860 saw the publication of his texts on Differential
Equations (1859) and Finite Differences (1860); the work involved in preparing
these appears to have taken much of his time in the years 1858—60 as no papers
were published in these years.
In the last years of Boole's life, 1862-4, his research activity seems to have
returned to differential equations - indeed, apart from the two papers on
AUGUSTUS DE MORGAN 3
probabilities mentioned above, all of Boole's papers in these years deal with
differential equations.
Boole died at the early age of 49 from pneumonia, the result of being
caught in an autumn shower. In spite of his life being spent out of the main-
stream of mathematical activity, Boole made a strong impression upon his
contemporaries. For example, Todhunter, writing in the introduction to his
History of Probability , refers to the interest shown by Boole in this work:
... by one prematurely lost to science, whose mathematical and metaphysical
genius, attested by his marvellous work on the Laws of Thought, led him nat-
urally and rightfully in that direction which Pascal and Leibnitz had marked
with unfailing lustre of their approbation; and who by his rare ability, his wide
attainments, and his attractive character, gained the affection and the reverence
of all who knew him. (Todhunter 1865, v)
Primary sources of information on Boole's life include: M.E. Boole 1878
(articles by Boole's wife); Harley 1866; Rhees 1955 (which gives comments on
Boole by contemporaries and pupils). An obiturary notice by John Ryall (a
colleague at Cork) appeared in the Illustrated London News of 21 January 1865,
59—61. The most useful modern accounts of Boole seem to be Kneale 1948;
Taylor 1956 (Taylor is a grandson of Boole).
Augustus De Morgan
De Morgan was born in 1 806, the son of an officer in the service of the East India
Company. After attending schools of little note in the west of England, he
entered Trinity College, Cambridge in 1823 and graduated B.A. in 1827. In
1828, when only 22 years old, he was appointed Professor of Mathematics at the
newly -founded London University — which institution was renamed University
College London in 1836. Apart from the period 1831-5, he occupied this post
until his resignation in 1866. In 1837 De Morgan married Sophia Elizabeth Frend
and they had three sons and two daughters. He died in 1871 .
De Morgan's interests and activities were extremely wide. During his under-
graduate years at Cambridge he was one of the group which became known as
the 'Cambridge Analytical School', whose aim it was to free the teaching of the
calculus at Cambridge from the out-dated fluxion terminology of Newton. His
participation in the work of this group is the earliest indication of a continuing
interest in education; indeed, this is, perhaps, the unifying feature of his widely
spread interests. He wrote on many aspects of education, not only on mathemat-
ical education; examples of the diversity of this interest include works on the
education of the deaf and dumb, and on the Ecole Polytechnique. His most
important contribution to mathematical education was, perhaps, the series of
mathematical texts he wrote — on arithmetic, algebra, trigonometry, and a large-
scale text on the calculus. Several of these appeared under the auspices of the
Society for the Diffusion of Useful Knowledge — the activites of which was
4 INTRODUCTION
another of his interests. He contributed many articles to the Society's Penny
Cyclopaedia.
De Morgan served on the council of the Astronomical Society for many years
and took the chair at the meeting at which the London Mathematical Society
was founded. He refused to be considered for a fellowship of the Royal Society
and for the award of an honorary degree by Edinburgh University.
Further interests included the advocacy of decimal currency, work as an
actuarial consultant, and book collecting. The last of these led to his very
thorough works of bibliography.
Now I turn to what may be considered his more important contributions to
scholarship. The best known of these are his contributions to the advancement
of logic. They began with an elementary text, First Notions of Logic (De Morgan
1839a), but it was not until 1846 that De Morgan began what was to be a life-
long series of papers and books in which he made notable advances in the ideas
and the symbolic representation of logic. These advances mostly centre around
generalizing the traditional theory of syllogistic reasoning. If their importance in
later times seems small, it is because Boole's entirely novel ideas resulted in logic
taking a quite new direction. The more substantial mathematical papers
De Morgan wrote are now almost wholly unknown; certainly he made no lasting
original contribution to mathematics, but his papers show an awareness of diffi-
culties and a generally sound critical approach to the received methods. In
particular, his papers on infinite series show him feeling his way towards the
ideas of a theory of divergent series, while those on the Foundation of Algebra
show him taking an approach which comes near to that of abstract algebra — an
approach which deduces algebraic theorems from a set of axioms and which was
not made explicit until 50 years later.
Primary sources of information on De Morgan's life include: S.E. De Morgan
1882 (his wife's biography of De Morgan); also the obituary notices in the
Athenaeum , vol. 50 , 25 March 1 87 1 , 369-70 ; and in the Monthly Notices of the
Royal Astronomical Society, February 1872, 1 12-18. A useful modern account
of De Morgan is given in Crowther 1968; and P. Heath's introduction to
De Morgan 1966, is helpful on De Morgan's work rather than for biographical
information. Rouse Ball 1915, although brief, is interesting.
Editing
Rather than presenting the matter of the letters according to the various topics
their authors consider, I have chosen to stick to a chronological arrangement.
This has the advantages of maintaining the integrity of each letter and of showing
the developing relationship between Boole and De Morgan. The division of the
ninety letters into seven Chapters created no problems as breaks in the corre-
spondence nearly always allowed this to be done naturally. It happens that the
chapters often show a predominating theme. Each chapter begins with a brief
NOTES 5
introduction which mentions some of the matter of the letters of more general
interest. Most chapters are then subdivided into a number of parts. Each part is
preceded by a commentary which deals with the particular matters raised in the
following letters. This leaves the letters themselves largely free from intrusive
comments. However on a few occasions a comment or translation has been
inserted (in square brackets) in the text of a letter when this seems the most
appropriate place for it. Minor individual points requiring elucidation are dealt
with in footnotes. In one case (in letter 10) a theorem is announced by Boole
which needs rather more extensive explanation; I have placed this in an
Appendix.
The transcriptions have been only slightly edited; misspellings and failure of
syntax have generally been left uncorrected and a few marks of punctuation
have been supplied where wanting. I have expanded some of the contractions to
improve the ease of reading; an unusual contraction used by Boole, ' = ns ' for
'equations', I have rendered in full. Initial and terminal greetings have been
omitted throughout. The dates have been given in a uniform fashion rather than
as the writers gave them and addresses have been omitted from the transcripts. 3
I have inserted in the text of the letters a reference to each indentifiable book
or paper mentioned: these references have the form B 1848a or D 18496, which
indicate items a, b under the years 1848, 1849 of the Boole, De Morgan entries
in the bibliography, respectively. For persons other than Boole or De Morgan the
reference is of the form Laplace 1812a, or simply as 1812a when this shortened
form is sufficiently clear.
Over one hundred persons are mentioned in the correspondence. I have
included some information on their lives and activities in the biographical notes.
Parts of a few of these letters have been published before in A. De Morgan,
On the Syllogism and Other Logical Writings (1966); the editor, P. Heath, quotes
substantial portions of the letters of 16 November 1861 and 21 November 1861,
together with briefer extracts from the letters of 13 July 1860 and 20 September
1862.
Notes
further information on the letters with their location may be found in the
bibliography.
2 S.E. De Morgan 1 882 (see the Bibliography for the full citation of references).
3 Up to September 1 849 Boole merely indicates his address as 'Lincoln'. From
November 1849 to June 1859, in general, he writes from 'Cork' or 'Queen's
College, Cork'. Then from June 1859 'Blackrock' or 'Blackrock near Cork'.
Exceptions occur when he is travelling, usually in the summer months; these are
indicated in the notes.
De Morgan lived in London and had four addresses in the period of the corre-
spondence. Up to February 1 845 he wrote from 69 Gower St. From that date to
June 1860 his address was 7 Camden St. Then he moved to 41 Chalcot Villas,
Adelaide Rd., N.W. From September 1862 the Board of Works ordered that this
address in future be known as 91 Adelaide Rd., N.W.
1
GETTING ACQUAINTED: 1842-1845
The years 1842—5 constituted the period when Boole's reputation was estab- -
lished. He had made contact with D.F. Gregory in Cambridge in 1839, and in
1840 four of his papers (Boole 1840 a, b, c, d) appeared in the Cambridge
Mathematical Journal In these years De Morgan had settled in the post of
Professor of Mathematics at University College London which he held for most
of his career.
According to De Morgan's wife, 'George Boole. . . had introduced himself in
the year 1842 to Mr De Morgan by a letter on the Differential and Integral
Calculus. . .' (S.E. De Morgan 1882, 165). The first letter of the correspondence
is De Morgan's reply to a letter from Boole which, as far as I am aware, has not
survived; this reply concerns difference equations, and refers to Laplace's
Theorie Analytique des Probability (Laplace 1812). In De Morgan's Differential
and Integral Calculus (1842a, 736—66) there is a discussion of difference
equations, and in particular a reference to Laplace: 'Such equations as the
preceding occur in the theory of probabilities and Laplace treated them by the
method of generating functions. . .' (1842a, 748). Thus it seems likely that
Letter 1 is De Morgan's reply to the letter with which Boole opened their
correspondence.
Letters 2 to 8, written between June 1843 and February 1845, all concern
the work that resulted in the publication of Boole's major paper On a General
Method in Analysis (Boole 1844a). Boole sends De Morgan an outline of its
contents in Letter 2, and asks advice on a suitable place for its publication.
Although expressing some modest doubts 'if only it possesses sufficient import-
ance', Boole is clearly concerned that the length of the paper may provide a
barrier to its acceptance, but says it would 'be very injudicious to divide it'.
After receiving a manuscript version of it De Morgan says he 'has read through
your paper with great satisfaction' in Letter 3 and in this letter as well as in
Letter 5 proceeds to give Boole detailed advice on the preparation of the manu-
script, and suggests that it should be offered to the Royal Society for publication
in the Philosophical Transactions. The Royal Society's acceptance of the paper
is recorded in Letter 6 of 28 June 1844. This paper resulted in Boole's being
awarded one of the two Royal Society's Royal Medals of 1844; this award is not
mentioned until considerably later in the correspondence (see Letter 37 of
August 1851).
8 GETTING ACQUAINTED: 1 84 2- 1 845
Letter 9, the last letter of this chapter, is the first of many in which Boole or
De Morgan acknowledge the receipt of a paper and comment on the ideas con-
tained therein. This beginning of a continuing exchange of thoughts on their
current work indicates that the relationship is changing from one in which the
relatively inexperienced Boole is deferring to the better established De Morgan,
to one in which the participants treat each other on a more equal basis.
The growing regard between Boole and De Morgan is indicated by the change
in the style of initial and terminal greetings of the early letters. In letters 1 and 2
they address each other as 'Sir'; in letters 3 and 4 as 'Dear Sir'; in all subsequent
letters they commence 'My dear Sir'. There is a similar growth in the warmth of
their terminal greetings: De Morgan uses 'Yours faithfully', 'Yours truly', 'I
remain, Dear Sir, yours faithfully', 'Yours very truly', in letters 1, 3, 5, and 8
respectively. While Boole, more humbly, writes 'Your obedient servant', 'Your
faithful and obliged servant', 'I remain your faithful servant', in letters 2, 4,
and 6 respectively, but changes to 'Believe me to remain, My dear Sir, Yours
sincerely' in 7 and 'Believe me, My dear Sir, Yours faithfully', in 9.
As the correspondence proceeds both eventually terminate their letters with
'Yours very truly' or some similar phrase.
Letters 1-9
The mathematical subjects raised in the letters of this chapter are mainly
concerned with differential and difference equations. These related topics were
one of the main areas of Boole's early research activity. Other topics mentioned
in connection with De Morgan's published papers (1844/, 1844&) concern
continued fractions and triple algebra.
In Letter 1 (29 Dec. 1842) De Morgan, quoting Laplace, refers to the differ-
ence equation
ay x + by x+1 + . . . +x(a'y x + b'y x+i + . . .)
+ x 2 (a"y x + b"y x+1 +...) + ... = 0.
In this equation x denotes an integral variable — today one would be more likely
to write such an equation in the form
Any n + Bny n+1 + C n y n+2 + . . . = 0,
using A n , B n , C n , . . . to denote polynomials in n. On the page that De Morgan
referred to Laplace commented: 'L'equation differentielle precedente n'est
int^grable generalement que dans le cas ou elle est du premier ordre, et alors les
coefficients de l'equation aux differences finies en y x ne renferment que la
premiere puissance de x,. . .' (Laplace 1812, third edition, 83). Thus here we
find Boole taking up a result which Laplace has stated in a general form, but
which he is able to solve only for quite special cases. The technique of solving
these equations described by Laplace as 'integration by generating functions' is
LETTERS 1-9 9
closely allied to the (later) Laplace transform method of solving differential
equations, and the work of Laplace under discussion (Laplace 1812) is one of
the sources of the Laplace transform. These ideas appear in Boole's major paper
On a General Method of Analysis (Boole 1844a), in Section D, pages 261—70,
where we find that Boole has, as De Morgan commented in Letter 1, 'extend [ed
the method] to a corresponding equation with two or more variables'.
In Letter 2 (19 June 1843) Boole gave De Morgan an outline of his ideas for
On a General Method of Analysis (Boole 1844a). When this paper, a long and
detailed one, finally appeared Boole had made considerable changes— he
remarked in Letter 7 (15 Jan. 1845): 'the paper is so much altered and the
applications so much extended that you will scarcely recognise it'. However, a
comparison of the summary with the published paper shows that the overall
structure of the paper was not changed; for example, the 'Inverse Applications'
of the summary correspond to Section D of the published paper. One change
that Boole did make was in the illustrative example given in the second section
of the summary where he gave the solution of the differential equation
, d 2 u du ,
x 2 — - 7 + x— + (n 2 + x)u = 0.
dx 2 dx
In the published paper we find (Boole 1844a, 239) that he has changed this to
d 2 u du
dx 2 dx
x — 2 +x— +(n 2 +x 2 )u = 0.
However, the case of (1) in which n = does appear in the paper (Boole 1844a,
237); the substitution z = log* reduces the equation to a Hill equation. 1
In Letter 5 (11 Dec. 1843) De Morgan's remarks relate mainly to notation,
but concern mathematics in so far as his notational preoccupation is here con-
nected with the operational approach to the definition of the fundamental
notions of the calculus. Although one approves of his stressing the importance of
making the notation quite clear, it is difficult to make much sense of his use of
the operational symbols as part of an argument supporting the case for carefully
defined notations. His first remarks '0'x is a specific [i.e. limiting?] case of
1 1 an <Kx + Ax) — d>x
\}j(x,Ax) or — — —
Ax
and would be better understood [why?] by a symbol like D Ax 4x than by
D x #x:' illustrate the common early 19th century British style of analysis - which
was already made obsolete by Cauchy's limit -based development of the calculus.
De Morgan was not unaware of this work, but could not divest himself of the
older ideas.
In Letter 6 (28 June 1844) Boole expresses his interest in a paper of
De Morgan's on continued fractions (1844/) and remarks that he had 'tried
1 GETTING ACQUAINTED: 1 842-1 845
unsuccessfully' to relate them to the theory of generating functions, but thought
a connection with linear functional equations was more likely.
De Morgan's paper concerns the calculation of the numerators and denomi-
nators of the approximations to an infinite continued fraction. Another paper
by De Morgan on continued fractions appeared in the Philosophical Magazine
(1844a). This concerns the reduction of a continued fraction
a bx_ ex
T+ 1 + 1 +' ec '
to a power series S^jc 1 . De Morgan shows how one may conveniently find the
coefficients P t in terms of the constants a,b,c. . . .
In Letter 9 (24 Feb. 1845) Boole indicated his interest in De Morgan's paper
on Triple Algebra. This is the fourth of a series of four papers with the general
title On the Foundation of Algebra (De Morgan 1839*, 1841a, 1843c, 1844ft).
Influenced by W.R. Hamilton's recent paper on quaternions (W.R. Hamilton
184%) De Morgan investigated triples of the form a% + brj + cf with the object
of defining operations of addition and multiplication under which they, like
quaternions, will conform as nearly as possible to the usual algebraic laws, i.e.
the field laws. For addition and subtraction the 'ordinary laws' were assumed.
If, further, the commutative and associative laws of multiplication and the
distributive law are assumed, and | is identified with the unit of arithmetic,
viz. 1 , then the structure of the system depends upon the products r? 2 , f 2 , and
7tf . De Morgan tries a number of possibilities, but recognizes that cases of the
failure of the associative law arise. He also examines the possible forms of the
'modulus of multiplication' - his term for the three-dimensional analogue of
y/(a 2 +b 2 ) in respect of a + ib. He realizes, as did W.R. Hamilton, that
V(a 2 +b 2 +c 2 ) will not serve; with one exception De Morgan's examples
yield square roots of a non-definite form in a, b, c for the modulus of multipli-
cation. Thus even had the associative law of multiplication not failed, he could
not achieve a field or even a division ring. At one point De Morgan exclaims
(despairingly?) 'I am not able to present any striking geometrical interpretations'
(1844ft, 147). 2
1. DE MORGAN TO BOOLE, 29 DEC. 1842
In reply to your's of the 23rd I have looked into Laplace (Th6orie des
Probab.) [1812] and cannot find that he actually integrates by generating
functions any equation with variable coefficients. He shows p. 82 how to
finding the generating function of y x when
ay x + by x+i + . . . + x(a'y x + b'y x+l + . . . )
+ x 2 (a"y x + b"y x+1 +...) + ... =
and his method would easily extend to a corresponding equation with two
or more variables. But, as far as I know, anything like an organized mode
LETTERS 1-9 11
of obtaining the generating function when the coefficients of the equation
are variable, must be new.
Laplace's problems do not require any variable coefficients: that is, he
does not treat any cases in which the chance of winning the xth game is a
function of the chance of winning preceding games, and ofx.
From your account of your method, I expect to be much interested by
it and hope you will soon be able to complete and publish it.
2. BOOLE TO DE MORGAN, 19 JUNE 1843
Some months ago I took the liberty of troubling you for a reference to
Laplace. In your reply for which it still remains to me to thank you, you
were pleased to express an interest in the subject of investigation alluded
to in my letter. I have now drawn up a paper embodying the principal
results of the inquiry which I have had some thoughts of laying before the
Royal Society. Before taking a step of this nature I am however anxious to
have the opinion of a more competent judge as to its propriety. Knowing
that you have written much on kindred subjects, shall I presume too far on
your courtesy in applying to you a second time? The following is a brief
analysis of the contents of the paper.
Direct Applications
Section 1st. Fundamental Theorem, being a general relation connecting
any linear differential equation or system of linear differential equations
with a corresponding equation or system of equations in finite differences.
2nd. Solution of linear and linear partial differential equations in series.
Systematic theory of such solutions has I believe never before been given,
the received method not providing for cases of exception & failure. Thus
in the equation,
d u du
— 5 + x —
dx dx
x 2 : ~ + x— + (n 2 + x)u =
of which the solution is
u = cosOj logjc)(a +fli* + a 2 x 2 + etc.)
+ sin(n logx)(Z> + b t x + b 2 x 2 + etc.)
a , b being arbitrary and in general
m(m + 4n )
, b m = etc.
the received theory fails unless the existence of the factors cos(« logx),
sin (n log x) be assumed. The same method would also fail as respects
equations involving a second number X not devlopable [sic] in ascending
powers oix. Every difficulty which can occur is provided for in the theory
which I have in this section given.
3rd. Finite solution of linear and linear partial differential equations.
The theory here developed is I believe the only one in which anything
12 GETTING ACQUAINTED: 1842-1845
approaching to a classification of the integrable forms of linear differential
equations is possible — the coefficients being rational functions of the
independent variables. So far as I can judge the method applies without
exception to all cases in which finite integration is possible. The form
under which the linear differential equation is treated throughout the
paper is
„ + 0i (±j e9 „ + ^JL) A + etc . = £ ,
It is on this equation that the reductions are effected and not on the scale
as in Mr Ellis's mode - the reductions are always effected at a single step
& not successively & in general by symbols of differentiation and not of
integration, some few cases excepted. 3 This section also contains an exten-
sion of the theory of equations with constant coefficients to a large class
of equations with variable coefficients.
Inverse Applications
1st. Theory of Series & Generating Functions
2nd. Application of the Theory of Linear Equations of Differences
3rd. Linear partial equations of Differences
To give an idea of the form of the analysis I subjoin a particular case of the
fundamental theorem as applied to Sec. 2 of the Inverse Method.
Employing m as independent variable let the equation of differences be
u m +0 1 (m)« m _ 1 +0 2 (w)« m - 2 +etc. =
then if u be the gen. function of u m so that u = z>u m x and if e = x the
equation for determining u will be
d ^ 3..j.jl /jL\„20 ..-!-.♦„ =^.4.^.^-1-/.-^
m + 0! — e w + 02 — e 2W « + etc. = c + c x e a + c 2 e™ + etc.,
Co. c i> c 2 etc - being arbitrary and equal in number to the order of the
degree of the original equation.
In the concluding chapter I have applied the theory of series to some
important expansions and to the theory of definite multiple integrals.
Some of the results to which it leads appear to be new & interesting. I may
add that the investigations [B 1843a, b] published under my name in the
two last Nos. of Cambridge [Mathematical] Journal arose out of this more
general inquiry although treated by a different method.
What has induced me to think that the Philosophical Transactions
would be a fitting medium for the publication of the paper (if only it
possesses sufficient importance) is that from the intimate connexions of
the different parts & from the constancy of references to the fundamental
theorem it could, I apprehend, be very injudicious to divide it into separate
portions. Should you have any suggestion to offer I shall receive it thank-
fully and with attention. If on any point I have not been sufficiently
explicit it is because as a stranger I feel that I may have already taken a
scarcely warranted freedom.
LETTERS 1-9 13
3. DE MORGAN TO BOOLE, 24 NOV. 1843 4
I have read through your paper with great satisfaction and return it to you
under a separate cover by this post.
With regard to the setting up, I see no remarks to be made but the
following.
d d
1. The compound symbol, — , — , will seriously augment the printing.
da dt
Would it not be better, both mathematically and typographically, if you
used D which might frequently be used by itself, with D e or D t for distinc-
d
tion when wanted. Do is line and lead — - is two lines and lead.
d6
2. May not the examples which lead to known results be abbreviated in the
work down to little more than statements of data & quaesita. I would not
have them left out. The longer expositions should certainly be kept when the
result is novel.
With regard to the manner of printing: I see no channel in this country
except the Phil. Trans, the Cambridge Phil. Trans, or the Cambr. Journal.
It is probably too long for the third & I am afraid Gregory is in no state to
attend to or decide upon it. Whether the R[oyal] S[ociety] would print it
or not is a question. I think they ought to do so, but in sending it to them
there is the nuisance of keeping a copy or employing someone to copy it
at their rooms as they are very dog-in-the-mangerish about what they call
their archives and will not return a paper even when they do not print it.
The Cambr. Soc. labour under want of funds and would look suspiciously
I suspect, upon anything long. I think if you do not mind copying it out
you should try the R.S. in the first instance. The Philosophical]
Mag[azine] I have no doubt would print a summary but it would be
decidedly too long for that periodical.
[P.S.] I have kept this by me to look at a point I wanted to see again.
1 Dec 1843.
4. BOOLE TO DE MORGAN, 8 DEC. 1843
For your kindness in examining my paper I can only express to you my
most sincere thanks. Your suggestion respecting a change of notation I
shall attend to should I have to print the paper on my own account or to
send it to a journal after its possible rejection by the RS. As however I
shall in the first instance lay the paper before the Society and as before
doing this I shall have to trouble a friend here to make a copy for me,
being too much engaged at present to transcribe it myself, I shall not be
able to make the alterations proposed in the copy presented to the Society.
Perhaps you will do me the favor 5 to give me the requisite address which I
have vainly sought in the Transactions.
You do not say whether the method which I have given for the solution
of differential equations by series is original otherwise than in form. I am
anxious to ascertain this point. If the method merely enables us to do
what there was an organized & general method of effecting before I should
not think the paper so far as respects this particular application worth
sending to the RS.
The postage on the MS having been rather heavy I beg that you will
permit me to return the amount in stamps.
14 GETTING ACQUAINTED: 1842-1845
5. DE MORGAN TO BOOLE, 11 DEC. 1843
My opinion of your paper is that the method is an original application of
the calculus of operations, and does in various instances effect, not what
could not have been done without it (for hardly any method in mathemat-
ics does that) but what certainly would not have been done without it.
Like all other methods it has its large classes of cases which it makes
practicable and easy, and which were not practicable and easy before. On
the score of newness of method, I think you need not hesitate a moment
to send it to the Royal Society.
If your friend's copy be (as usually happens with copies) more legible
than the original (not that there is any fault to find with that) it would
perhaps be better to send the copy than the original. In which case your
friend might be requested to substitute D for d/d0, d/dt and the like, and
you might, in reading it over, alter D into D , D t etc. where you think the
subscript symbol is wanted, which will, I think, happen seldom except in
enunciatory expressions.
I have rather an objection to D x , D y etc. in the calculus of operations,
unless x, y etc. be somewhat differently understood. If E denote the
direction to change x into x + 1 , then E h is that of changing x into x + h ;
accordingly
<b(x + h)-<t>x E h -1 ±
is . <px
h h
the limit of which is log E . <t>x, hence D = log(E). Now, trying the funda-
mental property of log E, we ought to have
log(E m ) + log(E") = log(E m+n ).
If by mere definition we allow
log(E m ) = mlogE
of course this equation is true: or if we use E m at the outset instead of E,
we find this true. But throughout it is requisite that (E — 1 )fh etc. should
be wholly independent of jc, and B x presents an appearance of depending
on x. I have some suspicion that D in the calculus of operations is not so
much d<j>x/dx as d0(x + h)/dh (h = 0). I am not very strong about this,
but I find it rather supported by the fact of the differential calculus being
rendered confused to beginners from being obliged to consider x as varying:
now in fact, we change x into x + Ax and make Ax vary: x itself is never
the variable, but stands for a specific value.
x is then a specific case of
,r a ^ 0(x + Ax)-0x
\jj(x,Ax) or
Ajc
And would be better understood by a symbol like D Ax .0x than by D x (/)x.
LETTERS 1-9 15
6. BOOLE TO DE MORGAN, 28 JUNE 1844
After all the trouble which I have given you with my paper I thought that
you would be interested to learn that the council of the RS have decided
to print it in the Transactions. I have adopted the notation which you
suggested to me and beg once more to thank you for the advice & assist-
ance which you so readily afforded me.
I was sorry to see so small a list of contributors to the last No. of the
[Cambridge Mathematical] Journal. Not that the No. itself suffered for if
in fewer it was in abler hands. 6
Yourpaper(asI suppose) 7 on continued fractions [D 1844/] interested
me much. I once tried unsuccessfully to discover a theory of continued
fractions analogous to the theory of generating functions. I suspect that
they will be found to be in reality connected with a certain class of linear
functional equations.
7. BOOLE TO DE MORGAN, 15 JAN. 1845
Will you oblige me by accepting the accompanying paper [B 1844a] . You
will perceive that I have adopted your suggestion relative to the notation.
In other respects the paper is so much altered and the applications so
much extended that you will scarcely recognize it.
8. DE MORGAN TO BOOLE, 6 FEB. 1845
I beg to thank you for your very valuable paper — which I hope to study
in a little while.
9. BOOLE TO DE MORGAN, 24 FEB. 1845
I ought to have sooner acknowledged your kindness in favouring me with
your memoirs. The one on Triple Algebra [D 18446] I have read with
great interest and quite agree with your views so far as I am acquainted
with them. I suppose that if there are beings who can conceive of space in
more than three dimensions the subject would have to them a more than
theoretical interest. The paper on Divergent Series [D 1844c] I hope to be
able to study shortly.
You may perhaps be interested in the following expansion remarkable
from its following after the first term the law of Taylor's series.
f \ x + T I = *>(*) + & (x) 7" + 7^ / °' (x) 7^ + etc -
\ ax I dx 1.2 dx
in which f (x) = e 1/2(d /dx } f(x). 9 To determine f (x) we must expand
the exponential and in applying the theorem we may suppose both sides to
operate on any function u. I may mention that the expansion was deduced
from that of f(ir + p) in my paper [B 1845a] and that I have sent it with
some others to the [Cambridge Mathematical] Journal though I should
think too late for the Feb. No. I notice it here because you are interested
in the Calculus of operations of which it appears to me to constitute one
of the most singular results.
16 GETTING ACQUAINTED: 1842-1845
Notes
^or the connections of this work in analysis with Boole's logical ideas see
Laital977.
2 For a discussion of the development of the concept of algebra in Britain at this
time see Richards 1980o.
3 'Ellis's mode' refers to Ellis 1840, 1841. Boole mentions these papers in Boole
1844a, 252.
4 See the postscript.
5 Here, and occasionally elsewhere, Boole writes '-or' where the normal English
usage was, and still is, to write '-our'.
6 Five of the twelve papers in the May 1844 part of Cambridge Mathematical
Journal were by the editor Ellis. The other 'abler hands' included A. Cayley and
De Morgan.
7 Initially the Cambridge Mathematical Journal did not print the name of the
author of a contribution.
8 Presumably this is De Morgan's reply to Letter 7, and the 'paper' is the one
referred to in the previous letter.
9 De Morgan writes e for the base of natural logarithms; this notation is not
uncommon in British books of this period.
2
MATHEMATICAL LOGIC AND IRELAND:
1847-1850
Almost two years intervene between the last letter of Chapter 1 and Letter 10
(8 Jan. 1847), the first of Chapter 2; this letter refers to a meeting between
Boole and De Morgan. S.E. De Morgan reports 'they did not often meet. . . but
when his [Boole's] visits did occur they were a real enjoyment to both — I believe
I may say to all, for I shared in pleasure of his conversation, ranging as it did
over a wide field of thought, and touching poetry and metaphysical as well as
mathematical science' (1882, 168). De Morgan rarely left London, so their
meetings could only occur when Boole had occasion to go there. Letter 10
indicates that Boole wished to read in the British Museum while in London. The
Museum's circular reading room was not yet built, and the accommodation for
readers was rather cramped; a writer in Bentley's Miscellany for 1852 drew a
pen -picture of the scene:
Every class of person haunts the place, from literary lawyer's clerk to the
revolutionary notorieties of Europe. There are hebdomadal humorists purloining
jokes, third-rate dramatists plundering plots, girls copying heads and flowers. . .
(Anon. 1852,529)
Letters 11, 12, and 13 (Nov. 1847) are of considerable importance as they
concern the influential books which Boole and De Morgan were engaged in
writing and publishing: Boole's The Mathematical Analysis of Logic and
De Morgan's Formal Logic. Both were published in November 1847. Letter 12
contains De Morgan's comparison of the algebraic symbolic approach to logic
which Boole introduced in his book, with De Morgan's own notation, which uses
symbols but cannot be properly called algebraic. In addition to the letter
De Morgan sent Boole, there is a draft which he wrote but did not send: it is
endorsed, in De Morgan's hand, 'De M to Boole, not sent'.
Boole's letters show the direction in which his research was tending. In Letter
15 (8 Dec. 1848) he says 'I have been quietly and steadily working at Logic. . .
And I believe too that I have also reduced the general theory to a perfectly
harmonious whole'. Letters 17 and 18 indicate that probabilities were entering
the field of Boole's research - a topic that will recur frequently in the letters of
1851. In Letter 20 (13 Aug. 1849) Boole says he has 'been applying the Logic
lately in some new fields and perceive nothing like failure or inconsistency'.
18 MATHEMATICAL LOGIC AND IRELAND: 1847-1850
From December 1848 several letters refer to Boole's application for the
professorial post at the newly -founded Queen's College, Cork. De Morgan sup-
ported Boole's application with a testimonial. S.E. De Morgan wrote: 'My
husband was, I believe, in some degree instrumental in obtaining the appointment
[for Boole] at Cork, where Sir Robert Kane, who had married our friend
Mr Baily's niece, was Principal' (1882, 168).
Letter 10
Letter 10 (8 Jan. 1847) is the first letter which contains a piece of mathematics in
detail. Boole writes for De Morgan 'a demonstration of the theorem which I
showed you this morning. . .' Boole addresses this letter from London where, it
appears, he was on a visit. The theorem claims that
J-oo \ x — Xi x — X 2 x — \ n J J-oo
where a t (l <i<ri) are positive. Boole published his result without proof in a
brief note in Liouville's Journal (Boole 184&0, and in a somewhat longer paper
in the Cambridge and Dublin Mathematical Journal (Boole 1849a). This paper
contains only very sketchy indication of proof, but there are several examples
and some development of the basic idea of the theorem. Thus the account of the
demonstration in his letter is the most detailed given by Boole. This proof
appears less than adequate in the eyes of a present-day reader. And indeed the
restrictions that must be made upon the function / (and upon the more general
substitution allowed in Boole 1849) are not sufficient. However the result is
substantially correct. An account of the result is given in the Appendix.
10. BOOLE TO DE MORGAN, 8 JAN. 1847 1
In accordance with my promise, I sit down to forward to you a demon-
stration of the theorem which I showed to you this morning, and which
you appeared to take an interest.
The theorem in question is
f dxfix x — ... r- -
J-oo \ x— \i x— X 2 x — \ n J J-
= f dxf{x). (1)
Consider first the equation
X t x — X 2 x — X„
(2)
in which at present we make no assumptions respecting the constants a x ,
a 2 , . . . ,a n . We have
LETTER 10 19
Ql <*2 a n _ A
x-v r •• r _0
X — Ai X — A 2 X — A„
and multiplying by the denominators and arranging with respect to the
powers of x,
x n+l -vx n -v'x"- 1 -v"x n - 2 . . . =
in which v , v ... are functions of v and of the constants a x , a 2 . . . a n ,
Ai , A 2 • • -An- Hence by the theory of equations
2x = v
Sdx = dv
2 having reference to all values of x which correspond to a given value of v.
Now by (2)
flx—^r V-~ fi T-|"^>- (4)
X A\ X Aj X A^
Multiply this equation by (3), and integrate we have
If/f*~ fi 7~ f3 r-~ 4 7-W- fA.)d.. (5)
J \ X— Ai X— A 2 X—A n J J
It follows from (2), that if we assume p and 4 as the respective lower and
upper limits of v, the corresponding lower limits of x, in the several
integrals in the first member of (5), will be given by the equation
a x a 2 a n
= V
x — \ x x — \ 2 x — -\„
and that the upper limits of x, in the same integrals, will be given by the
equation
a i <*2 a n
x . . . = q.
x Aj x A 2 x \ n
To this we may add the condition that to every value of v between the
limits of p and q there shall correspond real values of x. Of course the
upper limits of x may be permuted among themselves, and also the lower
limits among themselves, in an arbitrary manner.
Let p approximate to — °° (if in such a case we may use the term,
approximate) and q to °°. First the condition that to every real value of v
there shall correspond real values of x requires that v should not have a
maximum or minimum value. For, if v have a maximum value, any greater
value must make x impossible and if v have a minimum value, any less
value of V must render x impossible. Again that v may neither have a
maximum nor a minimum value for any real value of jc, the equation
20 MATHEMATICAL LOGIC AND IRELAND: 1 847-1 850
U-XO 2 (jc-X 2 ) 2 (*-X M ) 2
which is the criterion for determining a maximum or minimum value of v,
obtained by differentiating (2). Now these denominators of the several
terms in (6) being squares and therefore positive, the required impossi-
bility will be secured, by assuming a x ,a 2 . ■ .a n to be all positive, and it
will be obvious, that it cannot be secured in any other way. Therefore
a lf a 2 ■ . . a n must be regarded as positive.
Now as v approximates to — °° the values of x approximate to — °° or
exceed \ t , X 2 . . . X„ by some indefinitely small quantity 6 respectively.
We do not suppose 6 to be the same for each. Again as v approximates to
°°, the values of x approximate to °° or are less than X x , X 2 . . . X n by some
indefinitely small quantity 0. Hence (5) gives
(r+p'-'+f^V..
\J-oo J\ l+ e J\ 2 +e
+ r-~V/(*-- ^v --*v ••• -V) = r. d "w-
h n +e] J \ x-\i x-\ 2 x-\ n ) J -°°
Now 6 being indefinitely small, all the integrals in the first member, after
the first, vanish when the integrated function is not infinite within the
limits, and give rise, when it is so, the imaginary terms which Cauchy's rule
requires us to reject. In fact, integrals which become infinite within the
limits, are to be regarded as themselves the limits of more general integrals
which do not become infinite under the same circumstances. Hence in all
cases
{"4-^-^...^)=!"^) (7)
J -°° 1 x—Xi x—X 2 x—\ n ] ->-<*>
provided that a i ,a 2 . . . a n are positive.
In applying this theorem the rule above noticed must be observed, and
when either integral becomes infinite within the limits, the resulting
imaginary term must be rejected.
If the function f(x) be even, we have
f"4-^--^- •'•- Js r\ = 'i«* x * x > (8)
Jo \ jc— \i x-X 2 x-\ n ) Jo
The following are particular applications
ist rw* ! = [*<*'""* ' ■H"*^ ♦<"*"'>
Jo Jo J o
f
Jo
LETTER 10
dxe-<* 2+a2/ * 2) = e~*°Ld*e~
21
■J"
Jo
IT
1/2
e 2a which is known.
2nd
dx e * = dx e
rix-a/x)
Jo
dx e
.-<x -4<wc - 4a J /x 2 + a 4 /* 4 )
/. rdxe- ( * 4+a4 /* 4 >- 4a <* 2+ * 2 /* 2 > = e 6fl2 f ~d* e"* 4
J o Jo
(9)
and so on indefinitely.
In a similar manner were obtained the following
dx x
n-l/2
IXn-W 2
2\n
'o (a + bx + ex )
dx x
n- 3/2
r(n)c 1/2 (b + 2y/(ac)) n - 1/2
hW 2
(a + bx + cx 2 ) n r(n)a in (b + 2^/(ac)) n - 1/2
(10)
By a slight modification of the theorem, we can, from any definite integral
with any proposed limits, deduce an infinite number of other definite
integrals, having the same value. The transformations etc. may be repeated
or inverted, and it is obvious that in general the forms obtained will be
new.
I conceive that the most useful applications of this method will be
made in connection with the theory of multiple definite integrals. The
reductions of such integrals usually depends on integrations of the form
dx cos (0(jc))
J — oo
and it has only been for very particular forms of 0(x) that the reduction
has been possible. The above theorem enables us to vary the form of 0(*)
within the cosine, without affecting the dx without, with any new factor.
P.S. I wish to spend a few hours in the library of the British Museum
and a friend Mr Goodacre 2 would like to see the place. I am informed that
Sir H. Ellis cannot admit me without a recommendation from some official
person. Can you assist me?
You are at liberty to make any use of my paper that you may please.*
22 MATHEMATICAL LOGIC AND IRELAND: 1847-1850
Letters 11-13
Letters 1 1 and 13 form a prologue and epilogue to the more important letter 12.
In Letter 1 1 (3 1 May 1 847) De Morgan comments that he 'would rather not see
your investigations till my own are quite finished. . . you might have the same
fancy as myself. In a situation in which each was in the process of completing
an extended work on logic, De Morgan was being meticulous in ensuring that
neither could be placed in a position in which he could be accused of plagiarism.
Letter 13 (29 Nov. 1847) contains De Morgan's acknowledgment of the receipt
of a copy of Boole's book (1847a) and he suggests other persons to whom Boole
might send complimentary copies.
11. DE MORGAN TO BOOLE, 31 MAY 1847
I am much obliged to you for your note.
I had no objection to let Sir W[illiam] H[amilton] communicate
anything he pleased because I felt quite sure he could not look at logic in
any way that could give any view to a mathematician: and so I think it will
turn out. But you are another sort of person and I would much rather not
see your investigations till my own are quite finished; which they are not
yet for I get something new every day. When my sheets are printed, I will
ask for your publication: till then please not to send it. I expect that we
are more likely to have something in common than Sir W.H. and myself.
I should have sent my paper on syllogism [D 1846c] (the one already
published) to you by this post: but I remembered that you might have the
same fancy as myself - to complete your own first. Therefore when you
choose to have it, let me know.
In Letter 12 (28 November 1847) De Morgan made some detailed comparative
comments upon his book Formal Logic and Boole's Mathematical Analysis of
Logic which were published almost simultaneously. As well as this important
letter, we also have a draft letter which De Morgan wrote on the preceding day,
but apparently decided against sending; the draft is marked in De Morgan's hand
'De M to Boole not sent. Nov. 27, '47'. The draft has substantial differences
from Letter 12. To facilitate discussion of these differences I shall divide the
draft into three parts, calling them draft a, draft b and draft c.
Draft a consists of two paragraphs containing a general comparison of their
methods of writing logic; in Letter 12 these paragraphs were replaced by two
much briefer paragraphs. In draft b De Morgan takes one of Boole's problems
and solves it by the notation he used in Formal Logic, Letter 12 contains a
similar but clearer account. In draft c De Morgan takes up a number of particular
points concerning the techniques of Boole's symbolic notation; none of these
points were discussed in Letter 12. Thus draft a and draft c contain material
which add substantially to the content of Letter 12, while draft b adds little
additional information.
A further problem concerning the presentation of letter 12 and the draft
LETTERS 11-13 23
letter relates to the notation used by Boole and De Morgan in their newly-
published books; to assist the reader to understand this now unfamiliar notation
some explanation is necessary. The scheme adopted to present the transcripts
and accompanying explanation is therefore:
(i) A note of notations and terminology,
(ii) Draft a.
(iii) Letter 12.
(iv) Draft c.
(v) A version of the central part of Letter 12 in a notation that will be, we
trust, immediately intelligible to a reader familiar with present-day mathematical
logic.
Notations and terminology
The notation used by De Morgan rapidly became obsolete and will appear quite
strange. Boole's notation, however, is close to that of present-day Boolean
algebra.
Boole wrote 'x +y* to denote 'x or y\ but he used 'or' in the exclusive sense
of either-but-not-both. Boole wrote 'jcy' for 'x andj>' and '1 — x' to indicate the
contrary to x. So much is reasonably familiar; the main notation he used that
differs from present-day notation is his use of (usually) the letter v to denote
'some'. Thus
x = vy is read x is some y.
This notation enabled Boole to write all relations as equations. As examples,
consider
x = y(l-z) + z(l-y)
which might be written today as
x = (y &~z)v (z &~y).
Also x = vyz, which today could be expressed either as a statement about sets:
x C y n z ; or as a statement in the propositional calculus as x D y A z.
De Morgan used different symbolic notations at different periods; we are here
concerned only with the notation he used in Formal Logic (18476). In this work
he wrote x for the negation of X and
X) Y to denote every X is Y.
In addition in Letter 12 he writes:
X,y or {X,y} to denote Xoxy
Xy to denote Zand^.
De Morgan does not introduce quantifiers; consequently it is misleading to
24 MATHEMATICAL LOGIC AND IRELAND: 1 847-1850
read into his words present-day predicate calculus. However, he seems to be
aware of the difficulties that arise in the absence of quantifiers.
To illustrate De Morgan's style of writing one might observe his explanation
of contradictions in Formal Logic:
The pair 'Every X is Y' and 'some Xs are not IV are called contradictory: and so
are the pair 'No X is Y' and 'Some Xs and IV. Of each pair of contradictions
one must be true and one must be false: so that the affirmation of either is the
denial of the other, and the denial of either is the affirmation of the other.
(18476,59-60).
In both Letter 12 and draft c De Morgan refers to something as 'nonexistent':
in the third paragraph of Letter 12 'But zZ,yY are nonexistent', and in draft c a
similar remark, 'But they are nonexistent'. In his notation z, Z are contraries so zZ
is the conjunction of a proposition (or class) and its negation (or complement).
These remarks illustrate the manner in which Boole and De Morgan expressed
themselves when writing about sets; others write in this way included Schroder
and Husserl. They used an extensional language, considering the non-empty parts
of a set; thus the empty set of modern set theory is not present in their dis-
cussion. The idea of a universal set is, however, present in Boole's work. He used
the word 'a Universe' to denote 'every conceivable class of objects. . .' (Boole
1847a, 15).
A more recent discussion of this type of set theory was given by Lesniewski
who called it 'mereology' (Lesniewski 1930).
Draft a. De Morgan to Boole, not sent
I have to thank you for your paper on logic received this evening. My book
was published (publication meaning giving a copy in boards) on the 24th:
but if publication mean communicating printed sheets to a reviewer to
read, it was published some weeks ago. Some of our ideas run so near
together, that proof of the physical impossibility of either of us seeing the
other's work would be desirable to all those third parties who hold that,
where plagiarism is possible 1 = a wherever a is > 0.
My bookseller is to send you down a copy by the first opportunity.
If two individuals exchange a book on logic, is it simple conversion or
by contraposition. In talking of the things which are so nearly in
common I am not speaking of what I published in the Cambridge paper
[D 1846c], but of a doctrine of compound names and syllogisms, not
then evolved. I need not tell you that I am delighted with the views you
have given. My working processes are not so like those to common
algebra as to symbols, but more resemble the operations of our
heads.
12. DE MORGAN TO BOOLE, 28 NOV. 1847
I am much obliged to you for your tract [B 1847a], which I have read
with great admiration. I have told my publisher to send you a copy of my
logic [D 18476] which was published on Wednesday.
LETTERS 11-13
25
There are some remarkable similarities between us. Not that I have used
the connexion of algebraical laws with those of thought, but that I have
employed mechanical modes of making transitions, with a notation which
represents our head work.
For instance, to the notation of my Cambridge paper [1846c] I add
XY name of everything which is both X and Y
X, Y name of everything which is either X or Y.
Take your instance of p. 7 5
x = y(l-z) + z(l-y).
I express your data thus
1... X)Zy,Yz Zy,Yz)X...2 [fl
The following is all rule, helped by such perception as beginners have of
the rules which will succeed in solving an equation
From 1.
not X = x, etc.
{z,Y}{y,Z})x
zy,zZ,Yy,YZ)x
But zZ,yY are nonexistent
:. zy, YZ)x
or YZ)x or YZ)xY.
But from 2 yZ)X
or yZ)Xy
:. YZ,yZ)Xy,xY
:. Z)Xy,xY
[*]
But by 1 .
Xy)Zy, Yzy
XY)ZyY, Yz
or Xy)Zy
or XY) [De Morgan deleted this line]
by 2 x) {z, Y}{y, Z}
x)zy, ZY
xY)xyY, ZY
or xY)ZY
.\ Xy, xY)Zy, ZY
26 MATHEMATICAL LOGIC AND IRELAND: 1 847-1 850
or Xy,xY)Z [*]
or Xy, xY and Z are identical. [f]
[De Morgan drew curved lines from the text lines marked [*] to this final
line of the argument, [f] marks the portion of this letter for which a
modern version is given below. ]
This is far from having the elegance of yours; but your system is adapted
to identities, in mine an identity is two propositions. Perhaps I should pass
from
X)Zy, zY
to
Z)X, xY
more readily than you would. But I am not sure.
In fact there hang a multitude of points upon this question whether
complex or simple forms are to come first.
Draft c. De Morgan to Boole, not sent
The solution of the elective equations will, I have no doubt, be found
inexpugnable. With regard to the syllogistic process, there are unexplained
difficulties about v and about division by y. Here you have recourse to
verbal monitions about the meaning of v. The process of division is not per
se allowable.
xz = yz does not give x = y. Take page 35
y = vx
= zy
y x = vxzy admitted
= vxzy do.
Now you may separate
VZ.XY in my notation
No VZ is XY
But not No VZ is X
and yet VX.ZY give VX.Z
There is something to explain about the divsion by y.
I think with Mr Graves that y = vx is the primitive form. But v is not a
definite elective symbol, make it what you know it to be, and I think the
difficulty vanishes
y = yx
LETTERS 11-13 27
= zy
y X = zxy 2
= zxy
Now some Zs are not Xs, the ZYs. But they are nonexistent. You may say
that nonexistents are not Xs. A nonexistent horse is not even a horse; and,
(a fortiori?) not a cow. This is not suggested by your paper; but appears in
my system.
I see that must be treated as a magnitude in form y X 0/y is 0: but
0/y is not capable of interpretation.
In fact, your inverse symbol is not interpre table, except where use of the
direct symbol has preceded.
xy make a mark on all the 7s which are Xs
— (xy) Rub them out again
x
— (y) Rub out marks which never were made —
x
But I do not despair of seeing you give meaning to this new kind of
negative quantity.
It may be thus
on the other side as
= zxy
(xy)z =
is an equation of condition giving in my notation XY.Z or XY)z or Xz or
X:Z. But in the form (yz)x it is an identical equation, since yz = 0.
In (zx)y it is true also though no conclusion to a syllogism, since the
middle term is not eliminated.
Observe that the conclusion of the syllogism really is
Those Xs which are Ys are not Zs.
Quaere, is there not even another process of reasoning before we arrive at
the ordinary conclusion namely Those Xs which are Ys are not necessarily
allXs
Xs (not necessarily all) are not Zs
Or, is not syllogistic reasoning twofold in inference, on form and on
quantity.
A version of part of Letter 12
As the reader is unlikely to be familiar with De Morgan's notation, much
of the contents of Letter 12 may prove difficult to follow. To aid the
reader in following De Morgan's comparison of his symbolic notation with
that of Boole the following remarks are appended.
28 MATHEMATICAL LOGIC AND IRELAND: 1847-1850
To translate De Morgan's symbolism into modern symbols that preserve
his verbal expression, one can use the following lexicon:
in place of
put
which is read
X(F
XCY
every X is Y
X,Y
XV Y
XoiY
XY
X& Y
Xand Y
X
~X
not-X
Note that the symbol 'C is used in the set theory sense (rather than that
of the propositional calculus) as this preserves De Morgan's verbal ex-
pression viz. 'every Xis Y\
Translation of the part of letter 1 2 from [f ] to [f ] into this notation
gives:
i. . . x c (z & ~r) v (Y & ~z) (z & ~y) v (Y & ~z) c x. . . 2
The following is all rule, helped by such perceptions as beginners have of
the rules which will succeed in solving an equation
From 1 . Not X = x etc.
(~ZV r)&(~^VZ)C~X
(~z & ~F) v (~z & z) v (y & ~r) v (y & z) c ~x.
But ~Z & Z, ~F & y are non-existent [i.e. false]
.-. (~z«&~r)v(r&z)c~x
or r&ZC~X or F&ZC(~X)&r.
But from 2 ~r & Z C X
or ~r&zcx&~r
:. (y&z)v(~r&z)c(x&~r)v(~x«&r)
.-. zc(x&~F)v(~x«&r) [*]
But by 1
X&~7C(Z&~y)V(7&~Z&~10 or X&~YCZ&~Y
by 2 ~X C (~Z V y) & (~r or Z)
:. ~x c (~z & ~r) v (z & r>
~x & y c (~x & ~r & r) v (z & r)
or -X&YCZ&Y
:. (x & ~r) v (~x & Y) c (z & ~io v (z & r)
LETTERS 14-16 29
or (X&~Y)V(~X&Y)CZ [*]
or (X & ~y) V (X V Y) and Z are identical
[these final remarks being connected to the lines marked [*] by curved
lines in De Morgan's letter] .
13. DE MORGAN TO BOOLE, 29 NOV. 1847
I got your letter and the copy just now. As you know by this time I
received the other. I will give my second copy to Univ. Coll. Libr.
Pray send one to Dr Whewell — who takes great interest in such
things — and to Dr. Logan — St. Mary's Coll. — Oscott near
Birmingham — Also to Rev. Wm. Thomson, Queen's College Oxford — And
to (Mr Solly care of Mr Asher, Berlin) care of Mr Nutt Bookseller Fleet
Street.
Thomson and Solly are writers on the subject and all are real readers. 3
I find my publishers had not sent to you today — but they will
forthwith.
Letters 14-16
In Letter 14 (24 Aug. 1845) Boole thanks De Morgan for a copy of a paper on
partial differential equations (D 1848a) and comments: 'both methods are
ingenious, and the second especially so. . .'. Both of these methods involve
solving a partial differential equation by relating it to another equation or
system of equations. In the first method, which concerns a differential equation
of the form <l>(x,y, p, q) = 0,De Morgan 'contrives that this shall be the result of
elimination [of v] between A(x,y,p,q,v) = and B(x,y,p,q, v) = 0.' He
proceeds to form a system of six partial differential equations consisting of the
two latter equations and the four obtained by differentiating each of these with
respect to x and y. On elimination of p, q,r,s,t — note that as usual p, q denote
the partial derivatives bz/bx, bz/by while r, s, t denote b 2 z/bx 2 , b 2 z/bxby =
b 2 z/bybx, b 2 z/by 2 , respectively — from the six equations 'there will result an
equation. . . which will be more tractable than = 0.'
The second method concerns a partial differential equation of the form
4>(x,y, z,p,q,r,s,t) = 0.De Morgan forms the equation
t ~ s r
(j)\p,q,px+qy-z,x,y,- =- r,- j\ = 0.
He then remarks 'if either of these equations can be integrated, say by Z =
\p(X, Y), then the solution of the other is obtained by eliminating X, Y from x =
dZ/dX, y = dZ/dY, z=xX+yY-Z: De Morgan observed that Legendre had
used a special case of this procedure.
Letter 15 (8 Dec. 1848) shows that Boole had withdrawn his application but
had 'been induced to resume my application'. Boole's withdrawal may have been
in part due to his father's failing health. John Boole died four days after this
30 MATHEMATICAL LOGIC AND IRELAND: 1847-1850
letter was written. This letter also contains Boole's ideas on how he might
inform himself on methods of teaching at Cork should he succeed in obtaining
the post -- a fine illustration of the moral earnestness of the age.
In Letter 16 (3 April 1849) De Morgan makes some remarks which suggest he
had come close to the idea of three-valued logic. He says he had 'considered a
little the problem of — not name and contrary — . . . but any number of
names — a proposition in which the alternatives are more than X and not-X. . .
but never had the curiosity to investigate more than some simple cases of three
alternative. . .'. As far as I am aware he never discussed three-valued logic in his
books or papers; there is, however, a reference to modal propositions in 1847a,
232.
14. BOOLE TO DE MORGAN, 24 AUG. 1848
I am much obliged to you for your paper [D 1848a] on partial differential
equations which I have read with great interest. Both methods are
ingenious, and the second especially so: I hope to study them more fully
some time but for a long time past I have been quite unable to engage in
any mathematical pursuits. Nevertheless I rejoice to see that progress
continues to be made.
15. BOOLE TO DE MORGAN, 8 DEC. 1848
The Irish professorships with reference to one of which you were so good
as to give me a testimonial a year or two ago, are now about to be filled. I
had a short time since withdrawn my name from the list of candidates, but
I have been induced to resume my application. My hopes of success are
not very sanguine, although in one quarter I have lately met with all the
encouragement which the most generous friendship could suggest. Still I
do not disguise from myself that men equal to me in more attainments,
and possessed of other recommendations that I can lay any claim to may
enter the field of competition. Happily for myself, I feel that I can bear a
disappointment without either looking at myself as an injured man, or
taking fee with those pursuits from which I have already derived far more
real and solid gratification than any outward successes can afford.
However if one does resolve to enter the field it is the part of wisdom
to provide for the defence of weak points. And it has accordingly occurred
to me that it would do something to supply a defect in my claims and also
be right in itself that I should state my intention of spending some time,
(in the event of appointment), at one or more of the Universities, so as to
see the practical working of different systems of instruction. If circum-
stances should make it convenient for me to spend a week in London, may
I venture to ask you whether there would be any difficulty as to my seeing
something of the state of instruction in your own college - I do not mean
mathematical merely - though this is of course the most important to me.
I should give you no other trouble.
You will think me by this time almost lost to original investigations in
mathematics, so little have I lately done. But I have been quietly &
steadily working at Logic and I wish I could some time tell you a little of
LETTERS 17-26 31
the results of my inquiries. I believe that I have at length succeeded in
reducing all the mathematical applications to one general method more
comprehensive and yet more simple than those which I have published
and including them. And I believe too that I have also reduced the general
theory to a perfectly harmonious whole. These things give me the hope of
making the subject interesting and of giving to it a ready practical
value - ends which I conceive myself to have been very far from attaining
in my published Essays. I should think it very selfish to say all this, if I did
not know that you are really interested in such speculations.
Accept my apologies for thus troubling you.
16. DE MORGAN TO BOOLE, 3 APRIL 1849
The Irish question is not yet settled — I know.
I have considered a little the problem of — not name and contrary - X
and x, — but any number of names — a proposition in which the alterna-
tives are more than X and not-X. I looked at it enough to see the possi-
bility of wider classes of numerically definite distributions and logical
syllogisms arising therefrom - but I never had the curiosity to investigage
more than some simple case of three alternatives — I hope you will go on
with it.
I hope you will expand your view of probabilities — which I am not
sure I understand. I look for plenty of logical symbolization from you.
Letters 17-26
In these letters we see the relationship between Boole and De Morgan deepening.
Notice De Morgan's remarks in Letter 25 (8 June 1850) relating to the revised
proofs of one of his papers: 'if you find anything which has your own image
upon it — you must extract the evidence. . . that I may put the superscription
also'. De Morgan recognizes that he has been influenced by Boole's ideas, and is
anxious to acknowledge this. In the same letter 'I have two notes of
yours— always on hand for answer...'. The correspondence had, by 1850,
become a regular and fairly frequent practice.
There are two main themes in these letters: Letters 17—19 concern an essay
Boole had written on probabilities and Letters 20—24 and 26 concern Boole's
appointment to the professorship of Mathematics at Cork and his reactions on
taking up the post.
In Letter 17 (12 April 1849) Boole writes of sending 'the paper which I
hope you received', which, as the following remarks indicate, concerned prob-
abilities. This was, Boole said, a 'sketch. . . not designed for publication —but
was written mainly to register the actual state of my own knowledge. . .'. In the
next letter (21 April 1849) Boole sent De Morgan an appendix to this paper.
De Morgan returns the paper with some comments on 10 June 1849 (Letter 19).
Letters 17, 18 and 19 contain a number of remarks which give some
indication of the contents of this paper. The date of Letter 17 suggests that
the paper, which Boole refers to as a 'Sketch', was written early in 1849. The
32 MATHEMATICAL LOGIC AND IRELAND: 1 847-1 850
main theme is probabilities and there is some discussion of 'a first principle'. In
Letter 18 we read that Boole has sent 'a short Appendix. . . which exhibits the
application of the method to the syllogism'. In Letter 19 De Morgan, returning
Boole's paper, mentions 'the principle' as not his but coming from Laplace and
refers to an example concerning rain and thunder.
The Library of the Royal Society contains (among other Boole manuscripts)
an essay entitled: 'Sketch of a Theory and Method of Probabilities founded
upon the Calculus of Logic'. This essay is written in parts of two notebooks
marked (by Boole) '2 Logic; and '6 Logic'. An incomplete version of this essay
was printed by R. Rhees in Boole 1952, 141—66.
The essay is in two parts; the first consists of an explanation by Boole of his
symbolic approach to logic. This part is clearly intermediate in date between
Boole's The Calculus of Logic (Boole 1848a) — Boole refers to this paper at the
beginning of the essay — and Laws of Thought, 1854, as parts of the essay are
evidently an early version of certain topics of that work. However, Rhees suggests
(Boole 1952, 141 footnote), on the basis of a remark Boole made in Boole
1851/, that the essay must have been written in 1848 or 1849; for in the 1851
paper cited Boole says in a discussion of probabilities in relation to the observed
frequency of events: 'I shall present a solution to which that method conducted
me about two years ago. . .'.
The second part of the paper and the more important, in that it contains
entirely novel material, not a reworking of earlier ideas, is headed 'Of
Probabilities'. An important concern here is the question of independence of
events which, together, constitute a compound event. Boole quotes a 'Principle
W from Encyclopedia Metropolitana — the article in question was written by
De Morgan (D 1837c), and enunciates a principle of his own: 'The events,
whether simple or compound, whose probabilities are given by observation, are
to be regarded as independent of any but a logical connexion'. In the course of
the discusssion Boole introduces an example concerning thunder and rain.
The essay concludes with four appendices — Rhees only gives the first two of
these - the last of which has the title 'Of the probabilities of conditional events;
and of the syllogism'.
Thus each of the seven points detailed above which arise in the letters can be
linked with a similar piece of evidence relating to the notebook 'Sketch'. To sum
up this discussion: it may not be that it was the actual 'Sketch' printed by Rhees
which Boole sent to De Morgan, but it seems highly likely that it was a paper
which contained substantially the same ideas as those of the second part of the
'Sketch' together with one of the appendices.
Letters 22—4 refer to a visit Boole made to London in connection with the
Cork post and to a planned meeting with De Morgan which in the event did not
take place.
In Letter 24 (8 Nov. 1849), written shortly after Boole had taken up the
post, one notes his reactions to the sectarian divisions in Ireland — which still
LETTERS 17-26 33
persist in the Northern part of that island. Yet less than a year later we see in
Letter 26 (17 Oct. 1850) his disillusionment with this situation. The desire to
leave Cork he expresses here recurs several times during the succeeding years.
The Queen's Colleges were founded as a result of Peel's policy of social
reform in Ireland. Peel was the Chief Secretary for Ireland from 1812 to 1818,
and Prime Minister of the United Kingdom from 1841 to 1846.
Although the degrees of Trinity College Dublin were open to all, in practice
no Catholics attended Trinity College owing to its strongly Anglican atmosphere
and the expense of the education it provided. Scholarships and fellowships could
be held only by those subscribing to Anglican principles; religious tests were
abolished in 1873 however.
In 1845 a Bill was introduced to incorporate Queen's Colleges at Belfast,
Cork, and Galway as secular institutions, but allowed the various denominations
to provide pastoral care for their adherents. The Colleges opened in 1849; fees
were low and generous provision of scholarships was made. In 1850 the Queen's
University in Ireland was formed to provide a body which linked the three
Colleges.
Initially the secular character of the Colleges was unpopular with the
Anglicans and Presbyterians as well as the Catholics. The opposition of the first
two died away; that of the Catholics did not. The opposition of the Catholics
was led by William McHale, the (Catholic) Archbishop of Tuam. Papal rescripts
of 1847 and 1848 expressed disapproval of the Colleges and in 1850 an Episcopal
Synod issued a formal condemnation of them. Thus the number of Catholic
students who attended the Colleges was relatively small.
An unfortunate result of the creation of the Queen's College in Belfast is
referred to in Letter 21 (14 August 1849). The new College superseded the
Belfast Institution, and the appointment of J.R. Young, the professor of
mathematics at the Belfast Institution, to the new College was prevented by 'the
Presbyterian party who controlled the professors' nomination' (Dictionary of
National Biography, J.R. Young, vol. 63, 383). De Morgan tried to obtain a post
for Young (Graves 1882. 275-7, 283-4).
The later history of university education in Ireland has little bearing on the
correspondence. But it may, perhaps, be of some interest to note the foundation
in 1854 of the Catholic University in Dublin; J.H. Newman was the first Rector
of this institution, whose degrees were not accorded recognition. In the 1860s
further moves to evolve a university system acceptable to the Catholics were
made, but the main attempt to this end, the Bill introduced by Gladstone's
administration in 1873, was defeated; the government fell shortly afterwards.
(Beckett 1966, passim.)
Another topic which will recur in later letters is De Morgan's controversy
with Sir William Hamilton and his supporters, on the quantification of the
predicate; and De Morgan's series of lengthy papers On the Syllogism which
relate to this controversy. These matters are mentioned briefly in Letters 17
34 MATHEMATICAL LOGIC AND IRELAND: 1 847-1 850
(12 April 1849) and 25 (8 June 1850), but I shall defer comment on them until
later when they assume more substantial form.
Also in Letter 25 De Morgan refers to the use of two negatives in Greek. Both
Boole and De Morgan were competent in Greek, Latin, and French, and Boole
also in German; as later letters indicate, De Morgan had a slighter knowledge of
German. The general rule in Greek is that a negative followed by a simple negative
(i.e. ov or /Z77) denotes affirmation; but a negative followed by a compound
negative (e.g. ovdev) denotes a strengthened negative. The opening line of the
Orestes is an example of the second case; in A.S. Way's translation: 'Nothing
there is so terrible to tell' (see Euripides' Plays, Volume 2, p. 141, Everyman
library).
In the quotation from Aristotle, the situation is rather different; indeed, as
De Morgan remarks, Aristotle is 'making Greek'. For in the Loeb Classical
Library edition, due to H.P. Cooke, the phrase mentioned by De Morgan is given
as
ovk eoriv ov-binaux; ovu-avd poo-nos
(note the hypens), and translated:
not-man is not not-just.
Thus in this case there is a simple negative (the first ovk), the other negatives
being 'invented' compound words.
17. BOOLE TO DE MORGAN, 12 APRIL 1849
I have been spending Easter in the country & did not get your letter till
yesterday when I sent off the paper which I hope that you received. The
second example in the probabilities ought to stand first. There is, owing to
a peculiarity of form in all the logical equations which occur in the appli-
cation to probabilities, a somewhat shorter method of solution than the
one given in the Logic — but I did not think it necessary to notice it. I
believe there is also a general method of reducing the final algebraic system
but I have not completely examined this point.
I ought to mention that the sketch I send was not designed for publi-
cation — but was written mainly to register the actual state of my own
knowledge — and to serve as a record of what I had accomplished in the
event of my never accomplishing any larger design. As to the examples I
suppose that far better ones might be found — but I made those hastily for
a test of the method. I verified almost every result by independent con-
siderations as it was obtained — but unfortunately I did not note down the
steps of this process and what I have actually given to them in the text will
I fear be meagre.
I imagine that the principle which I have assumed (the independence of
the results of observation) must be regarded as a first principle. It is clear
that in any theory a first principle is needed, and I think that the hypoth-
esis of the independence of simple events is of this nature.
You need not return the paper until I ask for it.
LETTERS 17-26 35
P.S. I must not close my letter without thanking you for your kindness in
undertaking to look over the paper — of which I am very sensible.
18. BOOLE TO DE MORGAN, 21 APRIL 1849
I just write to enclose a short Appendix which I have written out today to
my paper & which exhibits the application of the method to the syllogism.
I think you have yourself somewhere remarked that the theory of the
probable syllogism is imperfect. If you have a difficulty in understanding
any part of the paper I should be happy to endeavour to remove it; & I
have a few more examples both in logic and probabilities worked out if
you would like to try the method on a new case yourself.
19. DE MORGAN TO BOOLE, 10 JUNE 1849
I return you your papers with many thanks.
To say how far I agree with you would be difficult at this time, as it is
my busiest time — and I read the paper two months ago — But I must urge
on you to continue and publish, for your mode of viewing the subject is
one which will serve those who disagree as well as those who agree. With
regard to the specific reference to my principle (I take this word from
Laplace, it is a very bad one) or rather the principle stated in Laplace, you
are to understand that it is merely the mathematical inversion of the
preceding one. It presumes, as does the preceding one, that the events are
known to be independent. In your instance of rain and thunder they are
known not to be independent. Laplace seems to have put this down to
prepare for cases in which ab and a might be more easily found than
b — so that qb/a would be the correct way of finding the latter - But I
cannot remember that such a thing ever happened.
20. BOOLE TO DE MORGAN, 13 AUG. 1849
I received last week the official announcement of my election to the
professorship of mathematics in Queen's College, Cork.
When I became a candidate for the appointment you were so good as to
give me a testimonial. I feel it right therefore to inform you of my success
and to say how much I am endebted to you for the assistance which you
so willingly rendered me. I shall at least endeavor to justify your good
opinion and kind wishes.
[P.S.] Let me take this opportunity of thanking you for looking over my
paper the receipt of which I forget whether I have acknowledged. I have
been applying the Logic lately in some new fields and perceive nothing like
failure or inconsistency.
21. DE MORGAN TO BOOLE, 14 AUG. 1849
I am very glad to hear that the electors have had the sense to accept your
offer of joining the Irish Colleges. Whether I am to congratulate you or
not, I cannot tell — for Ireland is a riddle altogether — I sincerely hope,
however, that by keeping out of their squabbles, you may be able to live in
peace.
36 MATHEMATICAL LOGIC AND IRELAND: 1 847-1 850
I believe you are better situated at Cork than you would have been in
the north of Ireland - At Belfast, poor Young whose writings you know I
have no doubt, is ruined by being left out - For the new Government
College destroys the Belfast Institution from which his means of living
came. I am assured that he kept out of the disputes of all parties — and
that he had therefore all parties against him.
I suspect you are likely enough to find that an appointment in a part of
the country where the Pope predominates will give you an easier berth
than you could have had among the Presbyterians Church people and
Papists mixed.
22. BOOLE TO DE MORGAN, 3 SEPT. 1849
I have to visit London on business connected with the Irish Colleges on
Friday next & shall probably remain in town some days. It would be a
great satisfaction to me to meet you & to have half an hour's conversation
with you while I am there. My residence will not be very far from either
Camden St or the University & if you will tell me whether you are likely
to be at liberty and when & where, I will make an effort to get to see you.
It is likely that I shall be engaged a good deal on Saturday & Monday but I
shall I suppose be at liberty in the evenings and certainly on the Sunday.
But if you will name your own time I will endeavour to make other
business bend to it.
Of course I only ask you this in the event of its really being convenient.
If you are engaged don't scruple to say so — as you have not a captious
person to do with.
23. DE MORGAN TO BOOLE, 4 SEPT. 1 849
I shall be very happy to see you on Saturday or Monday Evening - My
house if full of painters etc. and my family all away so that I cannot say
come to dinner - but I will have tea ready at 7 o'clock on the day on
which you inform me you can come.
You should however let me have your address in town as soon as you
know it - that I may be able to let you know if any thing happens to
change the Evening.
[De Morgan concludes the letter with a map showing how to reach
Camden St (which is near Mornington Crescent) from the northern end of
Tottenham Court Rd. The house no longer exists.]
24. BOOLE TO DE MORGAN, 8 NOV. 1849
Circumstances after all prevented me from paying you my intended visit. I
waited at home for a month expecting a summons to town which never
came unless to be followed by a speedy postponement. I am very sorry
that I did not get to see you. For many reasons I should have liked to meet
you.
I find myself very comfortable here [Cork] . At present everything
seems to promise harmony. I have met with nothing like intolerance
among the Roman Catholics with whom I have conversed. It is understood
here that the priests are favourable to our views but are withheld by the
LETTERS 17-26 37
peculiar position which the forward zeal of such bigots as MacHale and
O'Higgins has placed them in from manifesting their sympathies with us.
I have met with but one or two of the hierarchy myself but what I saw of
them confirmed this opinion which I had before heard expressed by large
numbers of their church. Indeed they have good reason to be satisfied.
Our statutes bind us from introducing problems on divinity into our
lectures (not that a professor of Mathematics, however sound a protestant,
would be likely to impugn the doctrine of transubstantiation, however
likely a chemist might be) and deans of residence have been appointed for
the three denominations. The bearing of the local authorities of the college
has been conciliatory in the extreme, more so indeed I think than was
called for. It were better to rest on the truth & justice of our principles
and leave them to make their way.
Judging from the mathematical examinations which are just over
elementary scientific education is in a low state here. I am desirous of
starting a class for schoolmasters. You have something of the kind in
connexion with your university. Could you give me any hints? or
information?
Mr De Vericour and I are in the same lodgings. He takes his cigar out of
his mouth to breathe out to you the kindest regards and souvenirs.
Following your advice and that of my friend [Charles?] Graves I intend
again to enter the lists on the side of Mathesis against the logicians.
25. DE MORGAN TO BOOLE, 8 JUNE 1850
I have two notes of yours — always on hand for answer — expecting daily
for many weeks past to answer by transmitting you the proofs which you
are to look over to please me & take care of yourself. But these proofs 4
have been delayed till now — and they are sent one by one — the paper
being I suppose longer than they like to set up at once —
Accordingly — as revises come — I shall send them to you one by
one — and you need not return them — if you find anything which has
your own image upon it — you must extract the evidence from the papers
I have seen & send it to me, that I may put the superscription also —
I have no particular news about either mathematics or logic — I do not
know whether you are aware that an English translation of the Port Royal
Logic [Arnauld 1850] was published a few months ago by
T.S. Baynes — who is, I understand, Sir W. Hamilton's locum tenens at
Edinburgh — and who is preparing a work on logic [Baynes 1850] in his
system. There was published two years ago at Oxford -by Mr Chretien of
Oriel College - a small octavo 'On logical method' [1848] which is an
interesting work - I mention these things — because I never heard of them
myself till the other day — but you may have more of logical acquaintance
& correspondence than I have. Do you know of anything written upon the
use of the negative in Greek — which may resolve this.
It is said, and justly, that two negatives, in ordinary Greek do not make
an affirmative — but a more emphatic negative. I remember no instance at
this moment but the opening of the Orestes
ovk korw ovbev bewov. . . - Nevertheless - in Aristotle - (De Interpr.
cap.X) two negatives make an affirmative and three negatives a negative — as
in ovk koTiv ov bucaux; ovk dpdpcoiroq- Was Aristotle here talking
Greek — or making Greek.
38 MATHEMATICAL LOGIC AND IRELAND: 1847-1850
You will find the solution of Barbara Celagrent 5 etc. in the
revises - when they come. I dare say I shall send one in a day or two.
When are your continuations to appear.
[PS] My kind remembrances to M. de Vericour - Tell me how you are
getting on in your College.
26. BOOLE TO DE MORGAN, 17 OCT. 1850
I think that you and I are sufficiently acquainted with each other to
justify me in asking you if you should hear of any situation in England
that would be likely to suit me to let me know of it. I am not terrified by
the storm of religious bigotry which is at this moment raging around us
here. I am not dissatisfied with my duties and I may venture to say that I
am on good terms with my colleagues and with my pupils. But I cannot
help entertaining a feeling to which perhaps I ought not to give expression
that recent events in this college have laid the foundation of a want of
mutual trust and confidence among us which would be to me far more
painful than any amount of outward hostility. For my own part I no
longer feel as if I could make this place my home. Perhaps this is a state of
feeling which I ought to endeavour to repress but it is not easy to do so. I
dread that the tone of our mutual intercourse and regard may hence-
forth be wanting in the cordiality and trust which seemed before to
prevail.
This is all that I can say to you on the subject at present but sincerely
do I pray that the anticipations which I have expressed may not be
realized.
Do not suppose that I have quarreled with any body here and am
anxious to get away on that account. In the affair of De Vericour I took
his part but temperately and maintained throughout a friendly corre-
spondence with the President [R. Kane] & Vice-President [J. Ryall] . It is
what I see around me and what I cannot but anticipate in the future which
causes me to think that I might consult my peace of mind and my real
utility in the world by quietly withdrawing to another sphere of labour.
Now this is what I would not say to any one in whose good feeling and
discretion I could not place entire confidence. What I ask of you is not to
mention these circumstances but to inform me at any future period of
what you suppose might suit me in England. No one else knows of my
present views and feelings.
Let me now turn from this subject and tell you that I am following
your advice and diligently preparing a work on Logic & Probabilities for
the press. Some of the most recent of my speculations in this direction
would I think interest you. There is a point at which my theory of the
Laws of Thought comes to bear on the question of Human Liberty - with
reference to the Intellect directly - and with reference to the Will by
analogy, and also be connexion with the former. My conclusion is that
there is a real phenomenon in the mind whether rightly called Liberty or
not which distinguishes it from the system of external Nature and which
admits of being as exactly defined by its properties as any other phenom-
enon. When the introductory chapter is printed you shall have a copy of it
and then if you care to see the others you may do so.
I hope Mrs De Morgan is well. Give my best regards to her.
LETTERS 17-26 39
Notes
*This letter is addressed from 19 Northumberland St, Strand. The Boole papers
in the library of the Royal Society contain a draft of this letter (1847 Boole
other manuscripts); the draft does not differ from Letter 10 in any significant
way.
2 1 have not identified Goodacre. Possibly he is either Robert or William Goodacre
who resided in Nottingham and wrote arithmetic texts.
3 T. Solly wrote two letters to De Morgan on logical matters in October and
November 1 847; these are in the library of University College, London, MS Add
97/5. 'Asher' is, perhaps, Adolphus Asher the Berlin bookseller who was used by
Sir A. Panizzi (the Head of the Department of Printed Books at the British
Museum) for the aquisition of German books for the Museum. St Mary's College
was a catholic college, where H.F.C. Logan, another correspondent of
De Morgan, taught.
4 The proofs are presumably those of the part of Transactions of the Cambridge
Philosophical Society, which contains On the Syllogism II.
5 'Celagrent' is an invention of De Morgan, analogous to Celarent. The letter g
following a vowel means that the premise (or conclusion) denoted by that vowel
takes the correlative copula: see De Morgan 1966, 54.
6 The 'affair of De Vericour' must have soon subsided; he was still Professor at
Queen's College in 1 864. 1 do not know what the controversy was about.
7 This letter was endorsed 'Private' by Boole.
8 The first time Boole sends Mrs De Morgan his regards. He does this only
occasionally before his marriage in 1856.
3
PROBABILITIES AND ECCENTRICITY:
1851
No letters that De Morgan wrote to Boole between June 1850 and January 1856
appear to have survived. However, the considerable number of letters that Boole
wrote to De Morgan in these years testify that the exchange of letters continued;
there were 16 in 1851 alone. Examination of the texts of these letters suggests
that De Morgan must have written at least ten letters to Boole in this year; some
of them may have been brief notes accompanying the reprints of De Morgan's
papers of which Boole acknowledged receipt. The lack of De Morgan's letters is a
matter for regret, but it does not seem to result in any serious lacunae in under-
standing of the matters raised by Boole.
There are two continuing topics in the letters of 1851. The more important
concerns probability. We have seen in the previous chapter that Boole had been
studying this subject. In Letter 32 (16 July) Boole gives a summing up of the
result of his researches: 'I am sure that no general theory of probabilities can be
established as any other than a preliminary general theory of Logic. . . I am sure
that a perfectly general theory may be established [which] I believe to be quite
beyond the scope or power of the received theory'. Clearly Boole was well on
the path to the publication of An Investigation into the Laws of Thought in
1854. Letters 33 to 37 (July -August) are all principally about one special
problem in the theory of probabilities: De Morgan's attempted solution to this
problem are evidently less than adequate; Boole points out errors in letters 35
and 36 (4, 1 1 August). The nature of this problem will be discussed later in the
chapter.
The less important continuing topic concerns John Walsh, of Cork, 1786-
1847, an eccentric who published a number of brief tracts claiming discoveries
which he thought superseded the usual calculus methods of treating problems
relating to curves. He also claimed to have discovered the general solution to
equations of the fifth degree. The references in Letters 27, 28, 30, and 39 to 41
concern Boole's memoir of Walsh (Boole 1851/i). This memoir is still worth
reading; it gives a good illustration of the 'quasi-insanity' (Boole's words) of an
amateur of science who is convinced he knows better than the experts.
De Morgan, not surprisingly, included Walsh in his collection of circle-squarers
and others whom he dissected in^l Budget of Paradoxes (De Morgan 1872).
LETTERS 27-30 41
By 1851 both Boole and De Morgan were near the height of their
powers - Boole was 36 and De Morgan 45. Many letters refer to the exchange of
their papers. But, as we shall see in the last letters of this chapter, there are also
a number of literary allusions.
Letters 27-30
Letters 27 (3 1 March) and 29 (6 May) contain some discussion which relates to
the meaning of the concept of a solution of a differential equation. In Letter 27,
replying to a query from De Morgan, Boole explains his understanding of 'a
primitive equation of f(x, y, (dyldx), (d 2 y(dx 2 )) = 0' - i.e. any equation
/i (x, y, (dy/dx)) = which yields f(x,y, (dy/dx), (d 2 y/dx 2 )) = on differen-
tiation. The last paragraph indicates that the question that De Morgan had put to
Boole concerned the distinction between a primitive of a first order equation
which is a general solution — which contains an arbitrary constant — and a
singular solution — which does not. De Morgan had discussed this matter in his
paper read in February 1851, On some points of the Integral Calculus
(De Morgan 1851ft), the first section of which is titled 'On the singular solution
of a differential equation of the first order'. In this paper he proposed a new
term, and a modified definition of a singular solution:
The singular solution of a differential equation has been usually defined as the
solution which is not any case of the general solution. In this paper I propose to
apply the term to any solution whatsoever, be it contained in the general
primitive or not, which results from any process that cannot introduce an
arbitrary constant: reserving the phrase extraneous solution to signify any
solution which is not a case of the general primitive (De Morgan 18516,
109).
Boole suggests, however, that 'we are not to wantonly meddle with definitions
which semper ubique et ab omnibus have been agreed upon.' The Latin tag here
misquoted is due to Vincent de Leans, who died about 450. In a work in which
he argued against Nestorian beliefs, Vincent de Lerins defined as catholic those
beliefs 'quod semper, quod ubique et quod ab omnibus' had been agreed upon
and accepted.
A minor scandal is referred to in Letter 28 (22 April) where Boole acknowl-
edges the receipt of a 'notice of Libri'. Guglielmo Libri who came to London in
1 848 under the accusation of having stolen valuable books and manuscripts from
the French libraries he visited in his capacity as Inspecteur de Bibliotheques.
De Morgan was one of a number of persons who thought him inno-
cent — Lord Brougham was another of his defenders — but his guilt is now
generally accepted. Libri, says S.E. De Morgan, 'became our attached and valued
friend, always recognising a firm and able defender in my husband, whose articles
in the Athenaeum and elsewhere were the means of establishing a belief in his
innocence in England' (S.E. De Morgan 1882, 177). A review of Libri's defence,
Lettre a M. de Falloux, etc., appeared in the Athenaeum, vol. 22 (May 1849),
42 PROBABILITIES AND ECCENTRICITY: 1851
484— 5. This review is unsigned, but may well be by De Morgan. There are several
brief notices of this controversy in the Athenaeum between 1848 and 1852.
De Morgan wrote an extended defence of Libri in Bentley's Miscellany for 1852
(De Morgan 1852g).
In Letter 29 (6 May) Boole again refers to De Morgan's paper (18516); he
considers that 'it appears to me to contain a true theory and I think an important
one in some respects.' He says that parts reminded him of ideas contained in a
paper of his own which 'has been completed for several months but is yet
unpublished.' Boole later published a paper On reciprocal methods in the
differential calculus — it appears in two parts — (Boole 1852, 1853) which seems
to be the one he is here referring to. The published paper concerns envelopes and
the elimination of constants in a set of equations.
The particular point at issue in part III of De Morgan's paper, and which
Boole refers to at the end of this letter, concerns the existence of solutions of
differential equations involving arbitrary functions. For example consider (with
De Morgan) the differential equation y" = 0; this has y = Hx 2 + Kx + L for
the general solution. Let / denote any differentiable function of three variables
u,v, w;if u,v, ware then made functions of a:,
dx dx dx dx
If, further, we substitute u =y", v = xy" —y, w = hx 2 y" —xy' +y then
^-f(u,v,w) =y'"(f u +xf v + kx 2 f w ) = 0;
dx
thus f{y", xy" — y',\x 2 y" — xy' +y) = is a solution of the differential
equation y'" = which, as Boole remarks, 'may be more general than y —
Hx 2 —KX — L = 0.' Of course, the point here is that if one takes the most
obvious integrals of y" — 0, viz. y" = c,y' = cx + d,y = \cx 2 + dx + e, then
this solution becomes f(c, d, e) = 0. Boole also says; 'There is need however of a
good deal of additional inquiry.'
In Letter 30 (28 May) Boole expresses his regret at having offended
J.J. Sylvester. In a postscript to his paper Sylvester had said 'my theorem on the
subject [Linear Transformations] , which is of a much more general character,
and includes Mr Boole's. . .' (Sylvester 1850, 281). Boole had responded that
Sylvester's theorem Is original in form only' (Boole 18516, 90) and had analysed
the difference between their results, concluding with the remark 'Mr Sylvester
has, I am assured, too much love for truth to feel offended. . .' (Boole 18516,
92). Sylvester's theorem concerned the relations holding between the coefficients
of a quadratic form in n variables and those of the form obtained when the given
quadratic form is subjected to a transformation in which r of the variables which
satisfy a set of r linear equations are eliminated. However, a 'Reply to Professor
Boole's observations. . .' (Sylvester 18516) shows that he was offended. Sylvester
LETTERS 27-30 43
describes Boole's remarks as 'extraordinary observations. . . which I cannot. . .
suffer to pass unchallenged', Boole next wrote a letter to the editor of the
Cambridge and Dublin MathematicalJoumal (Boole 185 Id) in which he tried to
conciliate Sylvester by saying that he wrote 'for the sake not of controversy but
of peace. . .' and 'I acknowledge that in the sense stated by Mr Sylvester. . . his
theorem is perfectly original'. He was, Boole said, 'convinced that the present
misunderstanding is simply the result of hasty judgement' (Boole 185 Id, 284—5).
In addition to these complicated mathematical questions, lesser matters
also attract their attention. In Letter 30 (28 May) Boole mentions — approv-
ingly — De Morgan's 'paper on Signs' (185 Id). Boole says: 'The views are in
essential points identical with some which I stated in a paper written many years
ago and sent to the [Cambridge] Journal while Gregory was editor but of which
he never acknowledged the receipt.' Gregory was the editor from its inception in
1839 until his death in 1 844, and his failing health was doubtless the cause of his
negligence.
De Morgan's paper, On the Mode of Using the Signs + and — in Plane
Geometry, is a good example of his ability to take an apparently quite pedestrian
elementary point of mathematics and illuminate it by explaining the various uses
of the signs + and —in geometry with exemplary clarity. De Morgan summarizes
these uses under ten headings which include the directions that arise in those
situations where axes are given, and of projections. Examples he gives include
AB + BC + CA = — where AB denotes the directed segment AB, etc; also
P°Q + Q°P = — where the notation P°Q denotes the angle the line P makes
with the line Q, etc. .
27. BOOLE TO DE MORGAN, 31 MARCH 1851
I have just now so much more to do than I can do well that I am unable to
give to your question the careful consideration which I would otherwise
most willingly do but I will just remark that it at present appears to me to
be a question of definition. If I define a primitive equation of
cbc d 2 y\
to be such an equation
/.|x,,,£|-0
as that the equation /= shall be a necessary consequence of the system
/i = 0, d/x/dx = [J then the test of f x =0 being a primitive of /= is
simply that the above condition shall be satisfied.
Now in the Universal Church of the Mathematicians to which you refer,
the famous canon of Vincentius Lirinensis applies thus far, viz. that we are
not wantonly to meddle with those definitions which semper ubique et ab
omnibus have been agreed upon and accepted. And this I hold to be sound
catholic doctrine.
44 PROBABILITIES AND ECCENTRICITY : 1 85 1
If therefore you press your question I call upon you, although protesting
against so unwarrantable an exercise of private judgment, to tell me what
new sense you put upon the term primitive equation. You must not tell
me that you mean 'ordinary or singular or both' but you must declare
what is that common property of the ordinary & singular in virtue of
which they are primitive.
[PS] Walsh in a few days.
28. BOOLE TO DE MORGAN, 22 APRIL 1851
I received your notice of Libri this morning and thank you for it. I will
show it to De Vericour. But I now write to ask if you have received my
account of Walsh which I forwarded to you about 10 days ago.
29. BOOLE TO DE MORGAN, 6 MAY 1851
I have looked at your paper [D 18516] with some care & without venturing
to give an absolute & final opinion upon the subject I may yet say that it
appears to me to contain a true theory and I think an important one in
some respects. I have met with other cases besides the general solutions of
partial differential equations of which your theory reminds me. They were
in connexion with the inverse problem of envelopes tangencies etc. of
which I have obtained a general theory introducing in certain cases
arbitrary functions of the constants of the problem. Now your functions
entering under the sign are functions of the differential coefficients
equivalent to constants. My paper has been completed for several months
but is yet unpublished and if you would like to see it I will send it to you.
I dont speak very positively about the analogy because I have not time to
set about studying the question in earnest just now, but it struck me on
reading & thinking over your letter. There is need however of a good deal
of additional inquiry. E.g. How is it that such equations as
f\y ,xy -y , — y ~xy +y\ - o
taken in all their generality admit only of the one original primitive as
their final integral an integral not involving any arbitrary functions or not
made more general by them. I don't mention this as an objection for it is
not one but it is a point worth looking into. Perhaps the solution
Xiy-Hx 2 -Kx-L) =
one of your solutions may be more general than
y-Hx 2 -Kx-L =
and have some meaning with reference to singular points etc which the
other has not.
All this is the mere result of impressions which the sending of your
letter has produced. There is a special business in which I am engaged for
our college just now that keeps me fully occupied in mind.
LETTERS 31-6 45
30. BOOLE TO DE MORGAN, 28 MAY 1851
I am now a little less busy than I was when I wrote to you last and will
take the opportunity of looking again at the differential equations and if
any thing new should occur to me I will write to you upon the sub-
ject — But at present I have nothing more to say than I said in my last.
Your paper on Signs [D 185 Id] has just reached me. I had seen it
before in the Cambridge Journal. I believe that the views are in essential
points identical with some which I stated in a paper written many years
ago and sent to the Journal while Gregory was editor but of which he
never acknowledged the receipt. I have not preserved even the notes of it
but remember getting in what seemed to me to be a very simple manner
some of the principal formulae relative to the rotation of a solid body. It
was in connexion with that problem that I had felt the necessity of fixing
with accuracy the sense of positive & negative with reference to rotations.
I think that I could recover some part of the matter if it were worth while
which as you have taken up the question I do not think that it is. Of
course I only mention this as a coincidence and not with the slightest view
to any personal claim. And indeed I should now feel precluded from
naming it any one but yourself. I would however advise you just to look
into the mechanical application. I think it led to theorems of transfor-
mation for rotations similar to the
x = ax' + by' + cz
etc. etc. etc.
where a = cos xx b = cosxy etc. and to some useful applications of them
dependent upon fixing the sense of rotation.* Something of this sort may
have been done since but I am not well read in these matters.
When you write again say what you mean to do with Walsh. Perhaps
you would think that I replied coldly to your proposal to publish it. But I
thought that you might have written under the immediate impression of
his strange story and that upon reflection your opinion might change. The
question is whether the publication would do good or whether it would be
interesting in a psychological point of view or not. Should you ever think
that there are sufficient grounds for its publication I shall willingly consent
to your doing so and shall not object to bear half the expense but I would
not have you proceed to publication unless you are tolerably clear about
the matter. A thing may be worth preserving which is not worth separate
publication. I am very sorry to have given such offence to Mr Sylvester. I
thought that I had not said one word that was not strictly true and even
called for by the mode in which he announced his theorem and I really
endeavoured to speak the truth in the manner least likely to wound him.
Letters 31—6
The mathematics contained in the letters of this section is primarily concerned
with probability and differential equations. Without question the most
E.g. the expression of pdt, qdt, rdt in term of d<J>, di//, dd Poisson Mecanique vol. II, p.134.
Poisson's reduction is very complex. [Boole's footnote]
46 PROBABILITIES AND ECCENTRICITY: 1 85 1
interesting matter is the discussion of Boole's general problem on probability in
Letters 33—7 (July— August).
Another point about probability arises in Letter 31 (24 June), viz. 'the general
doctrine among mathematicians concerning the probabilities of causes'. Boole
refers to two notes he contributed to the Philosophical Magazine in June and
August 1851 (Boole 1851/,g). These notes relate to Mitchell's problem of the
distribution of the fixed stars (Mitchell 1767). Also he refers to 'a passage in the
Edinburgh Review 1 - a review by J. Herschel of Quetelet 1846 which appeared
in volume 92 (1850), 1-57, of that journal. Mitchell's problem concerns the
question whether the observed distribution of stars is consistent with the
hypothesis of their being randomly distributed. The 'general doctrine' is explained
as follows: let p be the probability of the statement:
if the condition A has been satisfied, the event B has not happened. (1)
Now consider the statement:
if B has happened, the condition A has not been satisfied. (2)
The general doctrine, says Boole, asserts that the probability of (2) is/?: not so,
claims Boole, in fact the probability of (2) is
cjl-a)
c(\-a) + a<^-p) ,
where a is the probability of A being satisfied, and c the probability of B hap-
pening when A is not satisfied. The incorrect 'doctrine is explicitly maintained in
a passage in the Edinburgh Review' where it appears in the context of mineralogy
(page 32). In Letter 31 Boole also says the doctrine is 'I think strongly implied
by Laplace'. He gives detailed reference to Laplace's implied use of the doctrine
in Boole 185 Ig, where he quotes certain words appearing in the introduction of
Laplace's 'Great work on Probabilities', presumably Laplace 1812.
In present-day notation the problem is an easy one:
Pr(A'|B) = Pr(A' D B)/PrB = (Pr(B|A')/PrB
and PrB = Pr(B|A')PrA' + Pr(B|A)PrA.
Using p, c, a as defined above, Pr(A'|B) = c(l -a)/[c(l -a) + a(l -p)] , as
Boole says.
In Letter 31 (24 June) Boole is also at pains to make amends to De Morgan in
that he thought that the latter, too, had subscribed to the erroneous general
doctrine: 'I thought. . . it had your sanction but I find upon reconsidering the
passage. . . that it has not.' Boole exonerated De Morgan from error on this
matter in Boole ISSlg also. In Letter 35 (4 August) Boole again refers to this
paper, asking De Morgan to 'tell me whether you think that I have in any way
misunderstood you. . . and it is now surprising to me how I could ever have
mistaken your meaning.'
LETTERS 31-6 47
We turn now to the general problem which Boole stated in Letter 33 (24 July).
The problem concerns n events A,-, 1 < i < n, which occur with probability
c t , Ki<n; each of these events may be the cause of an event E, with prob-
ability p u 1 < i < n; required is the total probability of the event E. Boole asks
De Morgan: (i) is any solution known to him? (ii) does he 'think it can be solved
by. . . known methods or methods deducible from known science?' Finally
Boole remarks that 'when you answer this I will tell you more particularly why I
put the question to you.'
The reader conversant with modern probability may be asking whether the
events E f , 1 < i < n, are to be regarded as independent or not. Evidently
De Morgan made such a query in his reply for in the next letter (29 July) Boole
insists: 'I mean that there should be no restrictions but what are explicitly
stated. . . I apprehend that having given you the data it is not my business to give
you hypotheses.' The absence of any explicit statement concerning the indepen-
dence or dependence of the events E f , 1 < / < n, is crucial to understanding of
Boole's thoughts on this problem. Boole remarks in the Letter 35 (4 August).
'The grand difficulty in the common theory is to know what hypotheses you may
lawfully make and what you cannot' - i.e. precisely what relationship may hold
between E u 1 < i < n. In this letter Boole answers a letter from De Morgan con-
taining an attempted solution; Boole wrote: 'I am also obliged to you for your
solution. . . which however I think erroneous.' Its incorrectness is shown by Boole
by a counterexample which indicates that when n = 2, c x = c 2 =p, =p 2 = 1
the resulting probability of E is 4/3 !
It seems that De Morgan made a second attempt - but this gave Boole no
more satisfaction than the previous one. In Letter 36 (11 August) Boole again
writes out a statement of the problem with the comments: 'I think I should have
done better not to have endeavored to answer your questions. . . but simply to
have restated in clearer language the data and left you to analyse them yourself.
The case is simply this. . . You must I think admit that the data are clear and
intelligible. Don't you?' Again Boole is able to point out that De Morgan's
attempted solution is wrong, and he concludes with a strong statement that if
De Morgan cannot solve the problem either the 'ordinary principles' are insuf-
ficient or if they do suffice 'ordinary methods fail to direct us in their appli-
cations.' The final stage of these exchanges is reported in Letter 37 (25 August)
where Boole says he 'has decided to send it to one of the journals as it appears to
me to afford the most practicable and fair test which I know of the sufficiency of
the received methods in probability. When it has appeared you may wish to try
it again. If you do not I shall have no objection to communicate to you privately
and in confidence (which I would not do to every body) the solution.' The
surviving letters do not contain any indication of whether Boole did this,
however.
Boole sent his problem to the Cambridge and Dublin Mathematical Journal
where it appeared in November 1851 (Boole 1851e). He did not succeed in
48 PROBABILITIES AND ECCENTRICITY: 1 851
obtaining a solution, however. In a paper that appeared in the Philosophical
Magazine of January 1854 (Boole 1854ft), he stated: 'Several attempts at its
solution have been forwarded to me, all of them by mathematicians of great
eminence, all of them admitting of particular verification, yet differing from
each other and from the truth.' This paper is Boole's answer to 'the only
published [solution] I have seen' - by Arthur Cayley (Cayley 1853).
Boole and Cayley corresponded on the problem. The substance of their
exchange of ideas is contained in Cayley 1889, vol.2, 594-6. Apart from
references to Boole's and Cayley's papers, Cayley also, interestingly, refers to a
paper by Dedekind (1855). There are other references to the problem in Cayley
1862tf,6 (= Cayley 1889, vol.5, 80-85).
Boole presented his own solution to the problem he had unavailingly put to
his contemporaries in Laws of Thought. The problem with its solution appears as
Problem VI of Chapter XX (Boole 1854a). Boole mentions that the publication
of the problem failed to elicit a solution in Laws of Thought, although it 'led to
some interesting correspondence' (336, footnote). The attentive reader may have
noticed by now that Boole preferred to write 'the theory of probabilities' rather
than 'the theory of probability'.
31. BOOLE TO DE MORGAN, 24 JUNE 1851
I have been travelling about lately or I should have written to you before.
I send under another cover a paper [B 1851/] of mine which you have
probably seen before this time. I have sent to the same Journal a second
paper [1851#] in which I have corrected a misstatement made in the first
as to the general doctrine among mathematicians concerning the prob-
abilities of causes. The doctrine I spoke of is explicitly maintained in a
passage in the Edinburgh Review & is I think strongly implied by Laplace.
I thought too though I made no special references that it had your sanction
but I find upon reconsidering the passage at which I had looked that it has
not. I have therefore in my new paper stated what the views of writers are
on the subject so far as they are known to me and also what is in my
opinion a true summary of the theory which you will perceive should the
paper be printed to be partly taken from my former paper & partly
from your treatise.
On my return to Lincoln about the 28th or 30th inst. I shall send you
Walsh.
I hope you will look at my next paper [1851#?] & tell me if you agree
with me. I hope & believe that you will do so. I have discussed one or two
other points upon which from a former correspondence I think that we
differ but in speaking of these I have not alluded to our correspondence at
all.
32. BOOLE TO DE MORGAN, 16 JULY 1851
I should before now have answered your last letter if I had had anything
more than impressions to send you about the mathematics of your last
letter. But I have not. I think your views are just but I do not find that I
LETTERS 31-6 49
can at present so far disengage my mind from other things as to enable me
to study your demonstration with the care which would be necessary in
order to make my opinion of the least value.
What has occupied me lately has been the theory of probabilities at
which I have been working hard. The result is that I feel myself occupying
a ground upon which it is not presumptuous to say these two things: 1st
that I am sure that no general theory of probabilities can be established
upon any other than a preliminary general theory of Logic, 2ndly that
upon the principles stated in the little paper which I sent you in MS about
two years ago I am sure that a perfectly general theory may be established. 2
I now include in this the theory of observations, least squares, etc. etc. and
a great deal more which upon the most careful attention I believe to be
quite beyond the scope or power of the received theory. Of course I wish
you to retain the right of making every deduction for the infinite self-
deceptions of authorship but I have examined with care very case that I
have either been able to meet with or to think of and this is the conclusion
to which I am irresistibly led.
I am glad that you have seen the bishop of Cork. I think his a very
estimable person which is more than can be said for every body's bishop. I
am glad too that you have seen Sir R. Kane but I was previously aware
that you were acquainted.
[P.S.] If Taylor makes the least demur about Walsh do not urge him at all.
For he must be supposed to know best what will suit his readers & he may
say that he does not keep a psychological journal for chronicling the
delusions of a quasi-insanity.
33. BOOLE TO DE MORGAN, 24 JULY 1851
Are you too busy to give half an hour's attention to the following ques-
tion? If not you will confer on me an obligation.
You know well the solution of the problem in which are given the
probabilities of certain exclusive & exhaustive causes A x A 2 . . . A n and
the respective probabilities that an effect E will follow them taken singly,
and in which is required the total probability of the effect E. If c f is the
probability of the cause A,- and p { the probability that if that cause exists
the event E will follow then on the assumptions
Prob E = c x p x + c 2 p 2 ■ . • + c n p n .
Now first ; Is there any solution of the problem when the causes are not
assumed to be exclusive of each other? For example one observer attends
only to the cause A x and the effect E, neglecting all account of the other
causes and he finds that there is a probability c x that the cause A x will
take place and a prob p x that the event E will follow that cause. Another
observer attends similarly to the cause A 2 and consequent event and so on.
But nothing is known as to the connexion of the causes. All which is
known is that the event E cannot happen except from the causes singly or
conjoined. And what is required is the total probability of the event E as a
function of ci,c 2> . . . , c n , p lt p 2 , . . . , p n .
Secondly supposing that the problem has not been solved (I cannot
here refer for myself) do you think it can be solved by known methods or
50 PROBABILITIES AND ECCENTRICITY: 1 851
methods deducible from known science? I don't want you to spend much
time over it but I am very much interested to know what your impression
would be. I don't ask you this until I have anxiously considered the
subject myself. The result is that I cannot solve it except by my own
calculus. But I am not skilful in combinations, etc. I ought to add that the
result does not look as if it could be got in that way.
When you answer this I will tell you more particularly why I put the
question to you. You will see that I have a good reason for doing so.
P.S. I should have no objection to your asking any one else the question if
you know any one who has been working in this direction. I have asked
Mr Donkin.
34. BOOLE TO DE MORGAN, 19 JULY 185 1
I mean that there should be no restrictions but what are explicitly stated.
It might be better to use the word circumstances or events instead of
causes. Thus there are n events A] A 2 . . . A„ whose separate probabilities
are c x c 2 . . . c n also p t is the prob that if the event A,- happens the event E
will happen, whether by any causal efficacy in the event A,- or not. In fact
Pi is simply got by dividing the whole number of cases in which E has been
observed to be connected with A,- by the whole number of times in which
the event A,- has occurred. Further it is known that E cannot happen if all
the events A x A 2 . . . A n fail. What is the prob of E.
I apprehend that having given you the data it is not my business to give
you hypotheses. It is implied in the above statement that the events
At A 2 . . . A n are not mutually exclusive. There is no restriction on the
mode of their happening which is to be the most general possible consistent
with the values of c x c 2 . • • c n . The event E I regard not as quantitative
but as a simple phenomenon which either happens or does not happen.
I shall be greatly interested to know whether the above question is
really amenable to ordinary treatment. My own impression would be that
it is not.
P.S. You ask if one circumstance is regarded as hindering another when
simultaneous with it. I suppose the data do not imply any such hindrance
but the question is one which in applying my method I had no occasion to
consider.
35. BOOLE TO DE MORGAN, 4 AUG. 1851
I am much obliged to you for your paper which contains other matters I
perceive than were touched upon in our late correspondence. I am also
obliged to you for your solution of the question in probabilities which
however I think erroneous. For leaving z>i & v 2 unrestricted as you say at
the close of your letter we get
/o/o^'d^id^CciP! +c 2 p 2 -C!C 2 (Vi +V 2 ))
Prob = r i ,i-u. . ~,
J of o ' di>! dv 2
whence
Prob = CiPi + c 2 p 2 — \c x c 2 .
LETTERS 31-6 51
Now suppose c x = c 2 = 1, P\ = p 2 = 1, it is evident that the solution to
be 1 but your solution gives 1^ which is unmeaning.
The grand difficulty in the common theory is to know what hypotheses
you may lawfully make and what you cannot. And it is a difficulty which
in 99 cases of 100 is quite insuperable. I may just mention that Mr Donkin
gets
CiPi + c 2 p 2 -ofi
where a is the prob that both causes exist and |3 the prob that E exists if
both causes exist. But he remarks that there are difficulties in framing the
possible hypotheses as to the values of <x and /3 which he does not see how
to overcome. So far as he goes his solution appears to be consistent with
yours. To your process my objections would be 1st that you assume that E
whenever it occurs is distinctly referrible [sic] to one of the causes 2ndly
that you assume v x & v 2 to be equally susceptible of all values within the
limits of your integration.
I have thought of proposing the general problem which I regard as a
fundamental one to the consideration of mathematicians not at all as a
trial of personal skill but as a means of ascertaining the real power and
limitations of the received theory of probabilities. Of course I take upon
myself the responsibility for the correctness of my own views & of my
own solution.
I am anxious that you should read my paper [B 185 \g] in the Phil. Mag.
for this month & tell me whether you think that I have in any way
misunderstood you - for if I have I will endeavour to set all right. I really
thought when I wrote about the syllogism that you had maintained the
probability of the conclusion to be simply pq and omitted to notice the
possibility of its truth on other grounds; and it is now surprising to me
how I could ever have so mistaken your meaning. However I don't think
that this mistake appears in my paper although certainly if I had discovered
it sooner I should have omitted the passage altogether.
P.S. There is a copy of the Theorie Analytique des Probability of Laplace
advertised in Lumley's Catalogue for 12/0. It is the edition of 1820 avec
les trois supplements. Does it contain as much as the last edition. What is
the price of Poisson's work?
36. BOOLE TO DE MORGAN, 11 AUG. 1851
I think I should have done better not to have endeavoured to answer your
questions about the connexion of the elements in my problem but simply
to have restated in clearer language the data and left you to analyse them
yourself. The case you have to consider is simply this. The probability of
an event A x is c x that of an event A 2 is c 2 . The prob that if A! happens
an event E happens is p x and the prob. that if A 2 happens E happens is p 2 .
Finally E cannot happen unless one or both of the events A x A 2 happen.
Required the prob of E. You must I think admit that the data are clear
and intelligible. Don't you? Now I have nothing to add to them, - no
hypothesis to give you. You are from these data alone to solve this
problem on the principles of the theory of probabilities.
With reference to your first solution I have to remark that the limits of
52 PROBABILITIES AND ECCENTRICITY: 1851
integration which I employed were those given by yourself. With respect
to the second solution (received on Saturday) although I do not quite
understand the principle upon which it is obtained I can show that the
result is erroneous. E.g. Try p 2 = the others are not vanishing. 3 But it is
sufficient to ask you this question viz. whether the hypothesis which you
adopt and which you think that I in some of my explanations have
sanctioned is a legitimate consequence of the data stated above on the
principles of the mathematical theory of probabilities.
I have nowhere seen the fundamental positions of the theory better
stated than in your little book on probabilities [D 1838a] (Lardner's C.C.)
into which I looked for the first time the other day, and in a paper [ 185 1 ]
by Mr Donkin in the May no. of the Phil. Mag. Whenever the data of a
question are the probabilities of simple events there is no difficulty in
applying those principles. But when the data have reference to com-
pound events it is generally all but impossible to apply them without the
aid of a logical calculus. There are certainly no hypotheses involved in my
method if by hypotheses we mean something not necessitated by the
principles of the theory of probabilities and the fundamental laws of
reasoning.
If you do not solve the problem, as it stands, you must I conceive be
brought to one of these two alternatives: 1st that the problem in insoluble
on ordinary principles without inventing new hypotheses, in which case I
shall be at issue with you 2nd that if ordinary principles do suffice ordinary
methods fail to direct us in their application.
P.S. My best compliments to Mrs De Morgan.
Letters 37-41
The letters of the last months of 1851 are less mathematical in content, but
abound with references to literature. In Letter 37 (25 August) there is a mention
of the award of a Royal Medal of the Royal Society to Boole and in Letter 40
(17 November) of the publication of a lecture by him, The Claims of Science
(Boole 1851a).
In the nine years since their correspondence began Boole has developed a
freer and somewhat lighter epistolary style. This is particularly evident in Letter
38 (9 September), where Boole says: 'Mr Dickens has described the body of his
hero Quilp as finding its last resting place. . .'. Daniel Quilp, who is a villian
rather than a hero, appears in Dicken's The Old Curiosity Shop and Quilp's
death occurs in Chapter 67. Also Boole quotes the phrase 'fool'd by words' but
gives no further indication of the source of these words. This is perhaps a
recollection of Wordsworth's line
Slaves, vile as ever were befooled by words,
from no. XXIV of the Poems dedicated to National Independence & Liberty,
1837.
These literary allusions arise as a result of a curious misapprehension on
Boole's part. It appears that he did not know of the Kent coastal resort of
LETTERS 37-41 53
Broadstairs to which De Morgan had referred. The story of this mistake of
Boole's was told by De Morgan in a letter to W. Heald, 1 1 Sept., 1852:
'They [Mrs De Morgan and their children] have gone this year to
Heme Bay — not so far from London as last year, when they were at
Broadstairs. By the way a scientific friend of mine directed to me at
Broadstairs, near London, when near Ramsgate would have been nearer
the mark. On my asking him what he meant, he said he remembered some
very broad stairs down to the river just below London Bridge, and he had a
vague idea that they were the Broad stairs.
(S.E. De Morgan 1882,218)
In letter 41 (28 November) Boole thanks De Morgan for 'the very curious
paper about Newton and Leibnitz'. De Morgan wrote several papers on the con-
troversy concerning the question of priority between Newton and Leibniz on the
invention of the calculus. Indeed, De Morgan was the first British scholar to
point out the injustice done to Leibniz both by Newton's contemporaries and
later historians. To which paper Boole here refers is not clear; it may be either
On a point connected with the dispute between Keill and Leibnitz about the
invention of fluxions (1846ft), or On the Additions made to the Second Edition
of the Commercium Epistolicum (1 848c?). The latter is the more likely, except
that Boole acknowledges receipt of this explicitly in Letter 42 on 28 June 1852,
i.e. six months later. However, there is another possibility, as De Morgan's
contribution to the Companion to the Almanac for 1852 (1852/) entitled A
short account of some recent discoveries in England and Germany relative to the
controversy on the invention of fluxions would also be in print by this time. We
may note that Boole's first publication was an address to the Lincoln Mechanics
Institute: An address on the Genius and Discoveries of Sir Isaac Newton (Boole
1835) .
37. BOOLE TO DE MORGAN, 25 AUG. 1851
That you may be saved without knowing or believing any thing about the
distribution of the Royal Society's medals is a truth as indisputable as it is
consolatory. And therefore whatever popes and councils may say you
may relieve yourself of any apprehension upon that head. However to
gratify your curiosity (though I cannot but condemn it as a useless prying
into nonessentials) I have no objection to tell you that the RS did give me
a medal and further that Dr Mfarshall] H[all] s did once when I had
occasion to consult him tell me something of the kind which you mention
in your letter and offer to give me a full history of the whole affair either
then or at any future time with names etc. I declined the information with
thanks and should do so again were the offer repeated for although I was
grateful to the doctor for his good intentions towards me I thought that
the knowledge which I might get might make me think ill of some one of
whom I either thought well or thought not at all and so the gain would be
a loss. To poor Mr Davies I am grateful for the part he is said to have
taken. 5
54 PROBABILITIES AND ECCENTRICITY: 1 851
As to the question, I have decided upon sending it to one of the
Journals [B 185 le] as it appears to me to afford the most practicable and
fair test which I know of the sufficiency of the received methods in prob-
abilities. When it has appeared you may wish to try it again. If you do not
I shall have no objection to communicate to you privately and in confi-
dence (which I would not do to every body) the solution.
Hoping that you may enjoy your aquatic musings in which I am also
thinking of indulging myself.
38. BOOLE TO DE MORGAN, 9 SEPT. 1851
I have sent the proof [of Boole 185 \h] corrected to Taylor this day &
have taken upon myself to order 50 separate copies, 25 for myself & the
rest for you. If you do not care to have any will you write to the Editor &
say so. The cost for 50 is only 1/6 more than for 25.
I really did not know the locality of Broadstairs now so famous as 'our
watering place'. To tell you the truth I connected it in imagination with
some of those innumerable flights of broad stairs which one sees from the
steamers coming down to the Thames, and if I had any more particular
conception of its whereabouts it was that it was not far of from that
tidewash'd isle upon which Mr. Dickens has described the body of his hero
Quilp as finding its last resting place, somewhere between Thames a river
and Thames an estuary. Thus are we 'fool'd by words', in the first instance,
& left to the play of a vagrant imagination afterwards spite of all the
nonsense that is talked about Mathematics curbing the licence of fancy
etc. etc. etc.
39. BOOLE TO DE MORGAN, 10 SEPT. 1851
One of Walsh's titles of publications quoted in the memoir of him is
'Trinity College Notes of a Mathematical controversy between the
Rev. F. Sadleir Provost the Rev. Mr Luby and the Rev. Dr O'Brien Fellows
of the College and John Walsh Author of the Geometric Base' [Walsh.] . It
has struck me since sending off the proof yesterday that it might be
disagreeable to some of the above to have their names publicly mentioned.
There can no doubt of my perfect right to publish the title in full but I
am not equally clear whether it would precisely be doing as one would be
done by. And as I am connected with a University which is the rival of
theirs a more than ordinary carefulness to avoid giving pain is incumbent
upon me. Very likely I am only creating a shadow and I would therefore
ask you who know more of these matters to consider what would be
best. If an alteration is made it should be the substitution of an 'etc' for
all that follows 'Controversy'.
Will you if you think that there is ground for the omission forward this
note or its substance to Taylor immediately.
40. BOOLE TO DE MORGAN, 17 NOV. 1851
Will you oblige me by asking Messrs Taylor and Walton if they will print
and publish for me a lecture [The Claims of Science, B 1851a] which I
delivered at the opening of the College Session here and which some of
NOTES 55
those who heard it including my colleagues are desirous of seeing in print.
I publish the lecture at my own expense and fees here being few and small,
I should prefer that the lecture should be printed in a not very expensive
form. I took considerable pains over the matter of the lecture thinking
while I wrote it that it was possible I might be called upon to publish it,
beside which I felt as I have always done unwilling to labor [sic] for a
merely temporary object. This I mention that you may think that it is
solely on account of a few compliments that I design to publish the lecture
though at the same time it is possible that I may have misjudged the fitness
of the matter for publication.
I have been looking with interest at your paper [D 185 le] on evolutes
in space.
[P.S.] I directed the Ed. of the Phil. Mag. to send to me the account for
Walsh.
41. BOOLE TO DE MORGAN, 28 NOV. 1851
I have desired Taylor to send you the account for Walsh together with a
former 6/6 of mine due for another set. Will you settle both and then let
me know how much is due to you viz. the moity of Walsh with the 6/6.
Thank you for the very curious paper [D1848d] about Newton and
Leibnitz. I have sent my lecture to Taylor and Walton.
Notes
^his letter is addressed from: Mrs Knight's, 13 Surrey St., Strand. Perhaps
Boole had visited the Great Exhibition - which opened on 1 May
2 See Letter 17.
3 Boole inserted the sentence as a footnote.
'Lardner's C.C.' refers to Lardner's Cabinet Cyclopaedia, the 133 volumes of
which appeared between 1829 and 1849. De Morgan's Essay on Probabilities,
1838, was volume 107 of this work.
5 Boole is referring to the Royal medal he received for Boole 1844a. Boole first
wrote Marshall Hall, then crossed through all but the initial letters; however, the
name is still legible. Davies is probably T.S. Davies, 1795-1851.
4
THE LAWS OF THOUGHT AND MARRIAGE:
1852-1856
In this chapter all the letters but two are from Boole; only in 1856 do we have
letters from De Morgan again. One wonders why his letters from the earlier and
later periods should have survived, but none between June 1850 and January
1856.
Boole's activity in this period was centred around the preparation, and
publication in 1854, of his book An Investigation of the Laws of Thought
(Boole 1854a). In this work Boole recapitulates the ideas on the algebraic
formulation of logic put forward in his earlier book; also there is much on the
basic principles of the theory of probabilities. In Letter 42 (28 June 1852) we
read: 'I am now about to prepare for the press my long talked-of papers on
Logic and Probabilities'. And in Letter 44 (23 July 1852): 'I have something
like 500 pages in MS which however I am going to recast before publication'.
Letters 45, 46, and 47 (September-December 1852) of Boole refer briefly to
making arrangements for the publication of the book. Letter 49 (15 February
1854) contains a correction that Boole discovered was necessary just before
publication; he managed to get the correction printed in the book following the
table of contents. Letter 50 (23 February 1854) is evidently Boole's reply to a
congratulatory letter from De Morgan. De Morgan appears to have had some
doubts on certain points; but Boole's reply expresses notable confidence in the
correctness of his work: 'Satisfy yourself on this point - whether the solutions
my principle gives are ever false. If you find one instance in which they are I
give it up ... I don't think any man's mind ever was imbued with a more earnest
desire to find out the truth and say it and nothing else . . .'. Although we do not
have any letters from De Morgan to Boole at this time, De Morgan, writing to
W.R. Hamilton on 5 October 1852 referring to Boole, said: 'I shall be very glad
to see his work out, for he has, I think, got hold of the true connexion of algebra
and logic' (Graves 1882, vol. 3, 421-2.)
Again we find several letters in which Boole shows some dissatisfaction with
his position in Cork and his hopes of moving to a place which he may find more
remunerative and congenial. In Letter 51 (30 May 1854) he refers to financial
difficulties, but also says of Cork that he has 'become attached to the place and
to some of the people.' Nevertheless he considered the possibility of going to
LETTERS 42-50 57
Australia to take a foundation chair in the University of Melbourne. In Letter 54
(3 February 1855) he says: 'I am so out of the way here that all chance of
making any further advance is cut off unless I take some opportunity like this of
letting it be known that I should be glad to do more than I am doing'.
However, a change in his bachelor status was in the offing: in September
1855 Boole married Mary Everest. He met his future wife when she was visiting
her uncle, John Ryall, the Vice-president of Queen's College Cork. Another
uncle was Colonel Everest of the Indian Survey, after whom the mountain was
named.
Several letters throw light upon the personality and views of Boole. In Letter
42 (28 June 1852) we note his humanity in asking if De Morgan could help 'a
widow, struggling to bring up her children in London'. Letter 47 (8 December
1852) with his reference to the Hymn of Hildebert shows his breadth of
knowledge — the sacred poetry of the eleventh and twelfth centuries is surely
not reading matter that one would expect a mathematician and logician who was
self-educated to indulge in. Boole also refers to the Hymn of Hildebert in a
footnote of the last chapter of Laws of Thought, page 415. The footnote
indicates that the book in which he read this hymn was Trench 1849.
Letters 42-50
The mathematical topics raised in these letters are again mainly concerned with
probability. However, parts of Letters 42 and 48 indicate De Morgan's historical
interests.
In Letter 42 (28 June 1852) Boole thanks De Morgan for his paper on the
Commercium Epistolicum — the report of the committee set up by the Royal
Society to investigate the priority dispute regarding Newton and Leibniz and the
discovery of the calculus. The members of this committee were predominantly
Newton's friends and the report itself is strongly biassed in favour of Newton.
The first edition was published in 1712, and a second edition in 1722; the latter
was reprinted in 1725. The second edition purports to be a reprint of the first,
prefaced by some new matter. The substance of De Morgan's paper was that that
part which was ostensibly a reprint of the first edition had in fact been altered in
various places — De Morgan gives full details of the additions or changes - in
such a way as to make the anti-Leibniz bias more explicit and condemnatory. De
Morgan concludes: 'The more the whole matter is looked into from its beginning
to its end, the more will the evidence of reckless injustice thicken about the
enquirer.' Boole's comments show he had 'read a letter of his [Newton's] some
time ago giving advice to a young friend about to travel and I thought it full of
the spirit of a cold and calculating prudence.' No doubt Boole here refers to
Newton's letter of 18 May, 1669, probably to Aston; Boole would have read this
in Rigaud 1841, vol. 2, 292-5. It also appears in Newton 1959, 9-13. His
judgement 'cold and calculating' of this letter is most apt.
58 THE LAWS OF THOUGHT AND MARRIAGE: 1852-1856
Probability is discussed by Boole in Letters 43 (12 July 1852) and 44
(23 July 1852). In the first of these Boole answers a question raised by De Morgan
by referring him to 'the paper which I sent to you three years ago and which
contains all this part of the theory and a good deal more'. This paper must, I
think, be that referred to in Letters 17-19 and discussed in Chapter 2. But, as
Boole indicates in the last paragraph, he has improved upon the methods of that
paper: 'The modes of solution which I now employ are also considerably
different from those of the paper . . .'. The major part of this letter is an account
of 'an example of a solution got by me the other day'. This method was to form
a part of Chapter 19 of Laws of Thought (see in particular pages 316-18). In
brief, Boole's method was the following: there is a collection of events,
x t ,x 2 , . . . ,x n . Concerning these events Boole takes separately (i) the data - a
collection of statements of the probabilities of certain combinations of the
events; (ii) the quaesitum - the probability of a certain event which is expressed
in terms of x x ,x 2 , . . . ,x n . From (i) Boole derives what he calls 'the funda-
mental central equation', and he shows that there is a unique root, X, that
satisfies limitations imposed by the given data. From (ii) Boole derives a
function \p; the solution to the problem is then given by i//(X).
As an example Boole takes as data the statements: p t is the probability that
x t occurs, or all the events x t , x 2 , . . . , x n fail for 1< i < n. As quaesitum, the
probability of the event x t . He then shows (in summary in the letter, in more
detail in Chapter 19 of Laws of Thought) that the 'fundamental central equation'
is
1-X = (l-PiX)...(l-p„X) [3]
which he shows has only one (positive) root, X, satisfying 0<X<l/p 1 (he
assumes Pi >p 2 > . . ->P n )- The function \p is given by
0(X) = Pi-(l-c)p!P 2 ...p B X n_1 , [2]
where c is an arbitrary number satisfying < c < 1 - but, Boole adds, 'The
method gives the interpretation of c and informs us what new observations are
necessary to determine it'.
The type of problem considered by Boole - in particular the absence of any
explicit statement regarding dependence or independence of the events - has a
solution which generally contains one or more unknown parameters. Such
parameters may have to satisfy certain inequalities. Consequently, a modern
approach to such problems can usefully adopt the techniques of linear pro-
gramming. In recent work T. Halperin has examined Boole's logic and prob-
ability in this light (Halperin 1976).
The last eight lines of the first paragraph of letter 44 (23 July 1852) concern
what Boole was to call 'a perfect method' in Laws of Thought (Boole 1854a,
Ch. X). The substance of this method is contained in Proposition 1 of Chapter X
(1854a, 151): 'To reduce any equation among logical symbols to the form V— 0,
LETTERS 42-50 59
in which V satisfy the law of duality V{\ — V) = 0.' In the letter Boole observes
that any equations satisfying this law may be added together yielding an equation
which still satisfies the law. If the separate equations are v= 0, v — 0, v" = 0,
etc. and v, v , v" etc., satisfy the conditions v(l —v) = 0, v'(l — v') = 0, etc.,
then V — v + v + v" etc., also satisfies K(l — V) — 0; for we can write
V = v + (l- v)v + (1 - v){\ - v')v" + etc., (*)
and it is easy to see this satisfies V(l — V) = 0.
This technique used in writing V in the form (*) is exactly that used in set
theory when one has a denumerable collection of sets {At}T, and wishes to
express their union Uf A t as a union of pairwise disjoint sets: one writes
U?Ai = A, U (^.)U (A 3 \U\A t ) U (A*\(uUi) U . . .
The concluding paragraph of this letter contains an interesting remark: 'I have
long been trying to get at the principle of a suspected connexion between the
results of my methods and those of integration', which suggests that Boole may
have had some inkling of the common ground of probability and integration
which later became explicit in measure theory.
Letter 48 (7 February 1854) contains Boole's answer to an enquiry from
De Morgan about 'y° un g Murphy'. Robert Murphy, 1806—43, was the son of a
shoemaker in Mallow, who showed great ability in mathematics as a boy. A
Mr Mulcahy , a tutor of Cork, heard of his ability and money was raised to enable
him to go to Cambridge. He was 3rd Wrangler in 1829, and became Dean of his
college (Caius) in 1831. He fell into dissipated habits, left Cambridge in debt in
1832, afterwards living in London until his death in 1843. De Morgan wrote a
biographical notice of him in Volume 2 of the (first) supplement to the Penny
Cyclopaedia, 337—8.
42. BOOLE TO DE MORGAN, 28 JUNE 1852
I must thank you for your paper on the Commercium Epistolicum
[D 1 8 5 2d] which I have read. I am afraid that it is only upon an intel-
lectual throne that Newton must sit. I read a letter of his some time ago
giving advice to a young friend about to travel & I thought it full of the
spirit of a cold & calculating prudence.
I am now about to prepare for the press my long talked of papers on
Logic & Probabilities. The medium I have not fully determined upon. As
an application of Mathematics the probabilities will I think appear to you
the most remarkable that I have made. I was told a few days ago that a
work on the laws of thought (mathematical) has been presented to the
French Academy by a M. Courtois (or some such name). Do you know
anything about it! 1
I enclose a circular from a very deserving lady whom I have known
many years and who has recently been left a widow, struggling to bring up
her children in London. Should you think my recommendation sufficient
a word from you to any one residing in her neighbourhood might be of
service to her, & would oblige me.
60 THE LAWS OF THOUGHT AND MARRIAGE: 1 852-1 856
43. BOOLE TO DE MORGAN, 12 JULY 1852
The particular cases you discuss come under this general theorom stated
in my papers on probabilities viz. If 0(x y z) = 1 be the logical equation
expressing the occurrence of some particular combination of the events
x,y,z . . . and if p,q,r be the respective probabilities of these events
0(p q f) is the prob of the compound event above described.
I enclose you again the paper which I sent to you three years ago and
which contains all this part of the theory and a good deal more. You will
see from it some thing of the spirit of my method. It occurred to me when
you returned the paper that you had not had time to read it carefully.
Mind I don't want you to read it now, unless you care to do so but I think
it better at once to send you the paper which I do not want than to give a
necessarily more imperfect account of the theory as applied to your
examples by letter.
Here is an example of a solution got by me the other day. The following
particulars are known respecting n events Xi , x 2 . . . x n
1st The probability that either Xi occurs or all the events fail is p 1
2nd The prob that x 2 occurs or all fail is p 2 & so on
What is the prob that any particular combination 0Oi x 2 . . . x n ) of the
events X\ x 2 ■ ■ ■ x n will occur.
The solution is in all cases of the form
4> being a functional symbol dependent upon 0(xi x 2 ■ . . x n ) and known
when 00c ! x 2 . . . x n ) is given in form; X is a root of the equation
1-X = (l-PiX)(l-p 2 X)...(l-p„X)
Here you will ask how I know what root. The method itself tells me that it
must be that root of the above equation which is less than each of the
quantities — , — ... — , it further informs me that P\ p 2 . . . p n must be
Pi Pi Pn
subject to the condition
Pi +P2 ■ ■ -Pn > 1
in order that the problem may be a real one.
If the combination whose prob is required is the following viz. the
occurrence of some one and only one event of the series, the function \jj(K)
reduces to the form
PiX(l -p 2 X) . . .(1 -p„X) . . . + P W X(1 -PiX) ... (1 -p w -iX) 2
X
If the quaesitum is the prob of the event x t , i//(X) becomes
p 1 X-(l-c)p 1 p 2 [...]p n X"
X l J
c being an arbitrary constant admitting of any value from to 1. The
LETTERS 42-50 61
method gives the interpretation of c and informs us what new observations
are necessary to determine it.
I have proved that the equation
1-X = (1-PiX)...(1-P„X) [3]
has but one root which satisfies the conditions required when P\ Pt . . .p n
are fractions satisfying the condition
Pi +P2 • • -+Pn > 1-
This will give you an idea of what the method will do. There is always a
central fundamental equation [i.e. [3]] depending on the data alone &
independent of the quaesitum. The method gives this equation, & the
general form of the function of its root expressing the quaesitum, which
form becomes determinate when the nature of the quaesitum is known.
The method gives the conditions for limiting the root & it assigns the
requisite conditions among the constants of the data so that if probabilities
were given which experience could not furnish it would detect the im-
position. Finally it determines the nature of the experience necessary for
fixing the values of the arbitrary constants if there are any. It is applicable
to all sorts of problems.
By far the greatest difficulties I have had, have been in proving that the
algebraic equations for X have had one root and only one within the limits
assigned by the method. But I have always found that such is the case.
Now I don't way [sic] to trouble you in any egotistical way but this
seems the best answer to your letter. You will not find all that I have said
in the paper which I enclose but you will find enough to satisfy you of the
possibility of the higher matters which I have mentioned. The modes of
solution which I now employ are also considerably different from those in
the paper chiefly in this respect that I am able to dispense with arbitrary
constants c x c 2 & in the logical part of the solution & that the whole is
more symmetrical.
P.S. In apology for writing so late I must mention that I have been absent
on unavoidable business.
44. BOOLE TO DE MORGAN, 23 JULY 1852
Your letter followed me here 3 where I purpose to stay about a week
longer. I am glad that you propose to keep the paper, which I shall not
want again I believe, certainly not at present, and I would ask you to look
particularly at the rule of elimination which it contains for logical symbols
as well as at the probabilities, with this view that you may be able to
certify should it be required, that I was in possession of the rule some
three years ago. For it is really the turning point upon which all the higher
applications of the method depend. As to systems of propositions I will
only just say that my present practice is to reduce all the equations in a
system separately to the form V = in which V satisfies the law
V(l - V) =
and then any of the equations may be added together without arbitrary
62 THE LAWS OF THOUGHT AND MARRIAGE: 1 852-1 856
multipliers & the result will be equivalent to the equations thus added.
I agree with you fully that the laws of the symbols are independent in
some degree of the psychological views of different minds. Nevertheless
I feel convinced that my use of time is the right one. However I may get
up to London before long & if so I should like to talk to you about any of
these things. I have something like 500 pages in MS which however I am
going to recast before publication.
What you say about the origin of the [illegible] from the principle of
means is very curious. Something of the kind has occurred to me and I
have long been trying to get at the principle of a suspected connexion
between the results of my method & those in integration.
45. BOOLE TO DE MORGAN, 27 SEPT. 1852
I am going up to London tomorrow to make arrangements for the
publication of my book on Logic and Probabilities [B 1854a]. It has
occurred to me that you may be able to give me some information which
may be useful and accordingly I shall in a day or two probably write to
you again. 4
46. BOOLE TO DE MORGAN, 8 OCT. 1852
I do not remember leaving anything at my landlady's, 5 but think that
the book you mention may have been sent there for my sister by a friend
of hers without my knowledge. Perhaps you will be so good as to send it
& I will remit the cost. If you find it inconvenient to do so let it remain
for the present.
I set out for Ireland in a day or two. I feel doubtful whether I shall be
able to remain there long as I am never well when in Cork - the damp is so
excessive.
As to my book I shall in a week or two get estimates for it. There
would be some advantages in employing Gill I think. 6
Kant's argument from the Prolegomena is quite inapplicable. It is
certainly as impossible to prove the purely objective character of Space. It
is just like the old dispute about the reality of an external world. I do
think that when we know all the scientific laws of the mind we shall be in
a better position for a judgment on its metaphysical questions - of which
Kant's is one.
I will write soon after getting to Ireland & will not forget Mrs De
Morgan. 7
Let me thank you for the trouble you have taken.
47. BOOLE TO DE MORGAN, 8 DEC 1852
I am sorry that I have not been able conveniently to send Mrs De Morgan
before this time the Latin Hymn of Hildebert which I enclose - so much
of it at least (for it is very long) as is likely to interest her. It contains a
very good summary of the scholastic notions about the Deity in the first
portion, & the conclusion is really very beautiful. When I returned to Cork
I found that I had lent the book containing it to a friend and I did not like
immediately to ask for it. I ought perhaps to have written to say this but
LETTERS 42-50 63
Mrs De Morgan is I hope lenient to the failings of mathematicians - not
that I mean to imply that in the circle of domestic life she has need to
exercise this particular form of the great duty of charity.
I have agreed with Gill to print my book and hope to get a good deal of
the MS to press before the end of the year. I have chosen a tinted paper
some thing like yours and hope that our joint example may do something
to reform the public taste in this matter.
De Vericour tells me that he saw six months ago a notice of a report to
the Institute by Cournot on some paper on Mathematical logic. I mentioned
this to you 8 but did not remember then the name Cournot.
48. BOOLE TO DE MORGAN, 7 FEB. 1854
I cannot directly learn what you wish but believe it probable that Dr
Mulcahy's father who was a very able teacher of mathematics here dis-
covered the genius of young Murphy. I have however heard that one of the
fellows of T[rinity] C[ollege] D[ublin] dining with the rector of Mallow
examined him and gave him a college paper one of the more difficult
problems in which Murphy asked leave to take home but found out the
solution on his way and returned in haste and solved it. I can I think get
the name of the Fellow.
[P.S.] I think you pervert Shakspeare 9 who had certainly a soul above
proofsheets. And it is very well that he had for otherwise where would be
the glory of the critics who restore readings that never existed in his text &
of the philosophers who search out hidden meanings in his very mistakes. I
must further remark that one of your verses is defective in its number
which is not creditable to a professor of mathematics.
49. BOOLE TO DE MORGAN, 15 FEB. 1854
If you are reading my theory of probabilities [Laws of Thought,
B 1854a] I would wish you to interpret 'absolute probabilities' in Prop II
p. 261 as the probabilities which the events x,y, z ought to have in order
that if regarded as independent and as furnishing our only data the prob-
abilities of the same events under the condition assigned should be p, q, r
and interpret the problem of the urn accordingly. The solution of that
problem as it stands would be p = cp q = cq, c being the arbitrary
probability of the condition of a white marble, or white not-marble, or
marble not-white ball being drawn. The solution
, p+q—l , p+q— 1
p = q =
Q P
gives what the probabilities of a white and a marble ball ought to be in
order that regarded as independent and as our only data the probabilities
of the same event under the conditions should be p, q.
This does not affect the principle of the general demonstration in
Prop IV which is the following. By the logical reduction the solution of all
questions is reduced to a form in which the data are the probabilities of
simple events s, t . . . under a given condition V and the quaesitum the
prob, of a definite combination of those events under the same condition.
64 THE LAWS OF THOUGHT AND MARRIAGE: 1852-1 856
It is then affirmed that probability must be calculated as if the events s t
were independent and possessed of such probabilities as would cause the
probabilities of the same events under the condition V to be such as they
are assigned to be in the data.
This principle is certainly correct. I have seen its ramification in
hundreds of instances though it occurred to me in the first instance as
axiomatic and I hold it be so.
E.g. If the probabilitie s of a w hite ball w and a marble ball m under the
condition wm + w + m + 1 — wm are p, q what is the probability of wm
under the same condition.
Here Prob wm P Q
under the condition j p ' q > + p > { _ q > + ^ _ p '
p and q being determined by
P
pq +p \—q +q 1
i
Q
ii, i . i , i , i
+ p \-q +q 1 —p
= P
= q.
pq.
p+q-l , P+q-l
whence p = q =
and substituting
Prob wm under
the condition
= p + q-l
as is easily verified.
P.S. I have had a very kind letter of thanks from Sir J. Herschel.
50. BOOLE TO DE MORGAN, 23 FEB. 1854
I am much gratified with your letter and not surprised at the difficulty
you mention about the probabilities. But the principle to which you
object whether axiomatic or not is certainly true. You will be convinced of
this when you have read further on. You will also see that I have given the
general theory of quantification Chap XIX and further that it is connected
with the principle above mentioned — serving to determine the limits of
the roots of the equations furnished by that principle. I will write a short
tract on the laws of expectation & send you and I think remove your
objections. But at any rate satisfy your self on this point — whether the
solutions my principle gives are ever false. If you find one instance in
which they are I give it up. Are you satisfied with this declaration? I am
sure if there is any quality that I think you have in preeminence it is
integrity in the pursuit of truth — but that is a quality in which I should
be sorry to think myself your inferior. I don't think any man's mind ever
was imbued with a more earnest desire to find out the truth and say it and
nothing else, than mine was while writing that book. And the very
LETTERS 51-9 65
consciousness of this would make it not painful to me to give up half my
book if it were proved to be unfounded. However what I now ask of you
both as a friend of truth & of me is to examine the questions fully — to
settle it in your mind to make out whether I am right or wrong.
Do you now admit the validity of my theory of Secondary Propositions
and their connexion with Timet With sincerest thanks.
Letters 5 1-9
The matters raised in these letters are predominantly personal — concerning
Boole's thoughts of leaving Cork and his marriage. There are passing references
to mathematical topics including differential equations, the terminology relating
to the theory of invariants and a minor point of spherical trigonometry.
Letters 51 and 52 (May 1854) indicate that Boole had thoughts of leaving
Ireland to go to Australia. Melbourne University was set up by an act of the
Victorian Parliament of 1853. In January 1854 the Chancellor of the University,
Redmond Barry, who was the Puisne Judge in Victoria, wrote to a committee of
five persons, putting in their hands the task of selecting four foundation pro-
fessors. The committee included John Herschel, G.B. Airy, and R. Lowe. Barry's
letter included the following observations to aid the committee in their task of
selecting professors:
It is considered expedient that the persons whom you may deem eligible for
the office be men not in holy orders, of approved worth and moral standing, and
of such stability of character as to command respect. ... A devotedness on the
part of those selected to the cause of literature and the interests of the University
is deemed to be of great moment . . . And these general suggestions being sub-
mitted, it will be considered desirable that the Professors should be men under
the middle age, of approved diligence in literary pursuits, graduates of one of the
Universities of Oxford, Cambridge, London, Dublin, Edinburgh, or Glasgow, and
designated by some particular excellence in their collegiate career, accustomed,
if possible, to the inculcation of knowledge (with clearness and readiness) in the
department to which they propose to apply themselves, and, more especially, of
such habits and manners as to stamp on their future pupils the character of the
level, well-bred, English gentleman. (Melbourne 1854, 8, 9)
The committee received applications from 90 persons for the four chairs
(Scott 1936, 23), and 'on August 14th we agreed on Wilson, Rowe, McCoy and
Hearn' (Airy 1896, 220). The successful applicant for the chair of mathematics,
W.R. Wilson, was professor of mathematics at the Queen's College, Belfast. Two
other of the four foundation professors also came from the Queen's Colleges —
McCoy from Belfast and Hearn from Galway (Scott 1936, 23).
In Letter 52 we find that Boole had 'given up all thought of Melbourne'. It
is unfortunate that Boole forgot to put the date on this letter. However, a
comparison of the types of notepaper he used in 1854 and 1855 suggests that
the letter was written soon after the previous one. So no change of locale came
66 THE LAWS OF THOUGHT AND MARRIAGE: 1 852-1 856
about and he himself seems to have recognized that a move was unlikely. In
Letter 5 1 (30 May 1854) he says: 'I begin to feel that this is a wish of which it is
not the design of Providence that I should attain the fulfilment'.
In Letters 53 (3 January 1855) and 55 (21 February 1855) Boole refers to
some alternatives for the word 'determinant'. The word 'determinans' was
introduced by Gauss for the discriminant of a quadratic form in Disquisitiones
Arithmaticae (Gauss 1863, Bd. 1, 121). Boole 'had employed the term "final
derivative" for what has since been called the determinant' - in Boole 1843c
in fact. Boole mentions some terms introduced by Cayley in his work on
invariants. Cayley used 'hyperdeterminant' to denote an invariant in 1845, and
'hyperdeterminant derivative' in 1846. 'Quantic' was introduced by Cayley in
1854 to denote a homogeneous algebraic form.
This discussion of terminology is a reminder that Boole was one of the
originators of the theory of invariants. He wrote several papers on linear trans-
formations in which the idea of invariants arises (see Boole 1841, 1844c, 18516).
However, this early interest was one that he did not follow up later in his career.
In Letter 55 (21 February 1855) Boole mentions that he has 'used the word
eliminant for determinant' - as De Morgan had proposed in his paper (1854a).
However, Boole thinks 'there ought to be a better word from the Greek and I
will try to find one . . . Would not something having to "eliminate" or to some
equivalent verb the relation of -noirnua. [a construction, act or deed] to 7roieco
[to construct, make] be what we want.'
In Letter 53 (3 January 1855) we find one of the few references to public
affairs in the correspondence; although in Ireland in the aftermath of the famine
caused by the potato blight Boole never refers to it. In this letter, however,
Boole expresses his disquiet on the mismanagement of the British Army's
part in the campaign in the Crimea: at the time of writing this letter, January
1855 , Sebastopol had been under siege for several weeks. This war was the first
in which telegraphic communication made reporting of events on a day-to-day
basis possible, and so marks the beginning of the newspaper correspondent
sending regular reports from the battlefield.
In Letters 53 and 55 Boole refers briefly to a paper of De Morgan on dif-
ferential equations (De Morgan 1854a). In Letter 53 (3 January 1855) Boole
thanks De Morgan for the paper 'part of which I have read and of which I look
forward with interest to the further purusaf. However, in Letter 55 (21 February
1855) Boole says: T have been compelled for the present to stop in the reading
of your paper'. De Morgan's paper is, as he says himself at its beginning, 'of a
miscellaneous character' (1854a, 513), and is 40 pages long. Among other
matters De Morgan takes up again the points regarding primitives, singular, and
extraneous solutions which he discussed in his earlier paper (18516), and which
were raised in letters 17-19 of Chapter 3.
In Letter 55 Boole also remarks: 'Just now I am busy at analytical dynamics'.
He wrote few papers on applied mathematical topics and these could equally be
LETTERS 51-9 67
characterised as being on differential equations. In 1847 and 1856 he published
papers relating to Laplace's equation; the former (1847c) concerned the attrac-
tion of a solid of revolution, the latter (1856) was about the equation of con-
tinuity of an incompressible fluid. In this paper he refers to a letter he had
written to Charles Graves on this subject; the main idea is the use of quaternions
to obtain solutions to V 2 m = 0. Boole finds an appropriate form of Maclaurin's
expansion for a quaternion-valued function. Then he uses the factorization
bx 2 by 2 bz 2 " \bx +, by bz)\bx 7 by bz
where /, k are two of the non-real quaternion units i,j, k, to obtain a solution in
the form
1/2^ -L -i« f^\V2\ a -i/2 )-• a ^2 , ,. 90i
u = cos(jcA 1/2 )0! + sin(xA 1/2 )A _1/2 [f-r^ + k
by bz
where A = ^-y + —^ and <t> x , <f> 2 are arbitrary functions of y and z.
i!_ A 2
bx 2 + by-
Some years later Boole wrote a more substantial paper which he called 'On
the differential equations of dynamics' (1863a; abstract 1862c). This paper arose
out of an earlier paper (1862c) about simultaneous differential equations and in
its dynamical aspect relates to W.R. Hamilton's major paper which appeared in
two parts in the Philosophical Transactions in 1834-5 (Hamilton 1834).
Letters 56 to 59 (January— February 1856) form an exchange of correspon-
dence containing De Morgan's congratulations to the Booles' on their marriage,
and Boole's response. These letters show both writing in a humorous vein,
De Morgan making puns on mathematical terms and Boole in Letter 57 (8
January 1856) making 'one of such peculiar atrocity' that he has to explain it!
51. BOOLE TO DE MORGAN, 30 MAY 1854
I thought I might venture to ask you if you could tell me anything
(more than is contained in the pamphlet) about the Melbourne professor-
ships. I am in some doubt as to whether I should apply for one or not. To
speak candidly my income from the college has averaged scarcely more
than 300 £ per annum and as I have a mother & sister wholly dependent
upon me in England I see no prospect of making even the most moderate
provision for old age. Do you know if there are many applicants — if it is
likely that I should suit etc. I believe that I am pretty successful as a
lecturer & I have always been on the best of terms with the students.
I have now lived long enough in Cork to become attached to the place
& strongly so to some of the people. But I feel that I should not like to
spend the decline of life anywhere but in England. And I begin to fear that
this is a wish of which it is not the design of Providence that I should
attain the fulfilment.
68 THE LAWS OF THOUGHT AND MARRIAGE: 1852-1856
I will not ask you now to say whether you have made up your mind . . . 10
. . . that I have succeeded in verifying I think every point that was left in
the treatise conjectural or doubtful - the process leading in one instance
to a valuable extension (as I think it) of the theory of simultaneous
algebraic operations.
I hope Mrs De Morgan is well.
52. BOOLE TO DE MORGAN, NO DATE
I just write to say that I have given up all thought of Melbourne and to
thank you for your letter. If you should ever hear of any thing likely to
suit me in England I should be glad if you would let me know of it. My
objections to Ireland are however growing less and less and I have really
very little to complain of beside the smallness of the remuneration which
I receive. I incline to think that there are few places in Ireland so desirable
for residence as Cork and its environs. That the climate however diffuses
a kind of soft languor indisposing for exertion I feel sure.
53 . BOOLE TO DE MORGAN, 3 JAN. 1855
I am very much obliged to you for your paper [D 1854a] a part of
which I have read & of which I look forward with interest to the further
perusal. I do not in general think your papers easy to read but I think that
they repay the trouble of reading them. My eye fell upon a note in which
you speak of the word 'determinant'. I agree with you in your remarks. I
think the word was introduced by Mr Cayley when he took up the subject
of linear transformations. I had employed the term 'final derivative' for
what has since been called the determinant in that theory. But I quite
think that the word 'eliminant' which you propose would be better than
either. It may be doubtful whether the term hyperdeterminant could so
easily be replaced. However I think that eliminants covariants & invariants
would (with a proper liberty in the use of adjectives) answer every purpose.
The term Quantic recently introduced by Mr Cayley is in my opinion a
very bad one. It adds a Greek termination to a Latin adjective & expresses
nothing.
If you are still interested in the theory of probabilities you would I
think find advantage in looking at a paper of mine in the August No. of
the Phil. Mag. 'On the Conditions by which solutions are limited' etc.
[B 1854<2] and at some others in the same journal. I have however one to
send which will in my opinion put it beyond all dispute that the method
which I have published is a mathematically consistent one & that it is the
only such. The conditions of possible experience are identically the con-
ditions of success of the method viewed as an analytical instrument. I all
along felt that something of this kind must be the case but I have only
lately proved it after an investigation of extreme difficulty, which has
added some new theorems to the most difficult part of Algebra.
I don't write this to invite you to look at the paper unless it lies quite
within your plans to do so, and I shall not expect you to say anything
about it.
I have determined on applying for an examinership in Mathematics
under the commissioners for the affairs of India - not expecting to succeed
LETTERS 51-9 69
but feeling that I ought to make the effort to better my condition here.
Can you tell me when the elections are to be made & whether the
examinerships are to be tenable with other offices?
How melancholy is the intelligence from the east not disaster merely
but national humiliation & disgrace mourning that refuses to be comforted
in every household — sorrows in every heart! Want of patriotism seems to
me the radical evil. Members of parliament will not serve their country
without the bribe of patronage. Hence official incapacity, heads of an
army commissariat who do not know how to keep their charge from
starvation, registrars of colleges who cannot spell the English language and
graver abuses still. I have heard it said & partly believe it that a total
abandonment of all moral principles gives a man a power in official &
political life which nothing else will. Surely the end of these things must
come, and let it come. Order is not the first of the heavenly virtues.
I think this looks rather strong for a mathematician & yet I beg you to
think that I am not of a revolutionary spirit or a lover of change for its
own sake. But I have been profoundly impressed for a long time back with
the immorality of our official system in every department which I have
had the opportunity of examining. The loss of our whole army may
perhaps initiate a change.
11
54 . BOOLE TO DE MORGAN, 3 FEB. 1855
When I was an applicant for my present chair you were so good as to
give me a testimonial. I feel upon consideration a little scruple at using it
for another and different object without your permission. I therefore
enclose it wishing to say 1st that if you have any thoughts of becoming a
candidate yourself (in which case I should heartily rejoice to hear of your
success) I of course would not use it, and 2ndly if you have any objection
to my using it. If neither case should happen and you should desire to
make any alteration in the testimonial you are at liberty to do so.
As I said I do not expect to succeed but I think it a duty to offer
myself. I am so out of the way here that all chance of making any further
advance is cut off unless I take some opportunity like this of letting it be
known that I should be glad to do more than I am doing.
55. BOOLE TO DE MORGAN, 21 FEB. 1855
I have been so occupied lately that I forget whether I wrote to thank
you for the kind attention which you showed to my request & for the
addition which you further made to the testimonial. However if I did not
I do it now most heartily.
I have been compelled for the present to stop in the reading of your
paper [D 1854a] - only however for a time - as there is as [sic] a good
deal in it that I should wish to master. Just now I am busy at analytical
dynamics.
In a paper of mine which will appear in the Phil. Magazine for March
[B 18556] I have used the word ehminant for determinant, but it has
since occurred to me that there ought to be a better word from the Greek
& I will try to find one. The objection as it strikes me to ehminant is that
what it is meant to express is the result of elimination not a something by
70 THE LAWS OF THOUGHT AND MARRIAGE: 1852-1856
which we eliminate. It is perfectly true that the function ab —ab may be
looked upon as a canonical form by the aid of which we can eliminate x &
y from any equations of the form
ax + by =
ax + b'y =
so that in a cartain sense it might be said that ab' — ab is an eliminant by
which we can determine the result of elimination from any particular given
set of the requisite form as
x + py =
qx + y = 0.
But it may be replied that the relation of 1 — pq in the above instance to
ab' —ab is that of species to genus much rather than that of effect to
cause or work to instrument. Would not something having to 'eliminate' or
to some equivalent verb the relation of Tiov(\[ia to 7roieco be what we want.
The misfortune is that our verbs most of them come from Latin and that is
so miserably stunted a language in its verbal substantives that we cannot
get what we want.
The word 'determinant' is not liable to the same kind of objection in
one sense because it usually expresses a determining condition and in this
sense is active but it is subject to fatal objections in that 1st determining
conditions often are not expressed by determinants, 2ndly that the mathe-
matical origin of determinants is quite lost sight of. Don't think it necessary
to answer these hasty remarks. I will write again when I have read your
paper. [D 1854a]
56. DEM ORGAN TO BOOLE, 4 JAN. 1856
I happened to meet Mr & Mrs Stevens the other night, and the latter
informed me that you have been a married man some little time. If it had
been Mr S. I would not have given full credence, at once: but ladies are
always accurate on such points. I therefore confidently let fly a congratu-
lation, and beg you to present mine and Mrs De M's compliments to
Mrs Boole, and we hope that we shall have some opportunity of making
her acquaintance.
Of course mathematics and logic and probability have suffered for a
time - but no doubt they will raise their heads again.
I have been propounding a puzzle with nothing in it - You are to guess
an exceedingly elementary theorem of spherical trigonometry - not
thought worth insertion in any book - by the following enunciation
LETTERS 51-9 71
A beginner ought to know it - but where he is to get it from I do not
know. 12
57. BOOLE TO DE MORGAN, 8 JAN. 1856
My wife and I are both much obliged to you for your kind congratu-
lations. You see the information was perfectly correct. I have been a
married man now nearly four months. If wedding cards & the usual
ceremonies on such occasions had been observed you would have known
of the event at the time. You have heard doubtless of that division of our
sex into 'happy men' and 'lucky dogs' which some wit proposed to sub-
stitute for that of 'married men' and 'bachelors'. Well I have long felt that
the distinction was a real one and that to be a 'lucky dog' was not to be a
'happy man'. And this will in some degree explain my migration from the
one category to the other.
You sent me a little tract some time ago which I did not acknowledge -
but I did what was better. I read it through & liked it.
I don't at once see the way to your theorem in S[pherical] T trigon-
ometry] and I am afraid that I must leave it for the present as I am very
fully occupied. I only guess from an inspection of the figure that the
theorem is one of reciprocity in some way. Was this intended as anything
personal?
Thank you for your good wishes about the 'logic' and 'probability' in
connexion with my new state. I have only to say in reply that so far as I
can judge it is certain that the 'logic', and probable that the 'probability'
will not permanently suffer. Of course a man must as the old song advises
be 'off with the old loves - before he is on with the new'. 13 However I
don't see but that we may in this case continue to live together after a
while as a very 'happy family' and the more especially as my wife had
some previous acquaintance with her rivals.
We both reciprocate the kind expressions of Mrs De Morgan & hope
that her wish for a meeting may some day be realized. Present our best
regards to her.
P.S. Looking over my letter I see that I have described my migration from
the family of the 'lucky dogs' to that of the happy men as a removal from
one category to the other. You may know the pun was unintentional. It
has but just dawned upon me that it is a pun. But it is one of such peculiar
atrocity — it so closely resembles the murderous attempts of the country
correspondent of a Lincolnshire newspaper, that I protest against the mere
suspicion of having committed it by 'malice aforethought'.
58. DE MORGAN TO BOOLE, 13 JAN. 1856
If supplemental triangles had been called conjugate, you might have
made out a case of personal allusion. As it is, you cannot.
I imagine how various persons will smile when they receive the solution
of my riddle — and will say, is that alii It is as follows.
Nobody has, so far as I can find, taken the trouble to state the way in
which acuteness of sides and angles is connected with obtuseness. I can not
find such a thing in Cagnoli, Delambre, Puissant, Legendre or T.S. Davies. 14
I shall be glad if you can tell me where the following is found.
72 THE LAWS OF THOUGHT AND MARRIAGE: 1 852-1 856
In any spherical triangle (excluding the intermediate case where there is
a right angle or quadrant) — either such side is of the same name (acute or
obtuse) with its opposite angle, or — some odd number of acute sides is
joined with some odd number of obtuse angles and conversely, every case
just described exists, except three acute sides with three obtuse angles.
Now all this is seen in the diagram I gave
where the lower numbers mean numbers of obtuse angles the upper ones
mean numbers of acute sides. Then
means that one obtuse angle (meaning only one) may coexist with 3, 2 or
1 acute sides. But
means that all sides acute cannot coexist with more than one obtuse angle.
That reciprocity which you interpret into personal reflexion upon your
3
present condition — is that of the supplemental triangles — the and at
3
the beginning and end being supplemental etc.
I suppose if I had spoken of differential equations you would have
interpreted the general solution as the husband, and the singular solution
as the wife, and the contact of the latter with all cases of the former as a
reflexion upon the constancy of woman & variableness of man, or as a hint
that the former is up to every dodge of the latter - or something like that.
You know the derivation of the words husband and wife? They are
from the Sanscrit which compresses a good deal into a few letters. The
word wife originally means a demanding of money — and the word husband
means a person who deceives himself and the truth is not in him if he
imagine that by any possible method he will avoid forking out.
I dare not send compliments to Mrs Boole after the last fling.
59. BOOLE TO DE MORGAN, 23 FEB. 1856
I am your debtor both for a pretty little theorem in Spherical trigon-
ometry and for a paper of a more important character. And I have nothing
NOTES 73
to send you in return but thanks. The paper I have about half read and
with great pleasure and I should have read the whole if I had not been
obliged by conscience and [illegible] to stick to a paper of my own which
is long behind its time and which had been
short of an oath to be ready at Xmas last.
short now. My wife who has forgiven the Sanscrit roots sends her kind
regards to you and Mrs De Morgan.
1 promised by almost every thing
t. s For the same reason I will be
Notes
1 The person Boole named as Courtois was in fact Cournot; see Letter 47.
The numbers on the right of the formulae are editorial insertions. In equations
[ 1 ] and [2] Boole in error wrote 1 — X in place of X in the denominator. I have
corrected this slip in the text.
3 Boole addresses this letter from: Wickwar Rectory, Near Wotton Underedge,
Gloucestershire. The rector, the Rev. T.R. Everest, was Boole's father-in-law.
The village's name is more often written Wotton-under-Edge.
This letter does not make very good sense as it stands. Perhaps Boole intended
to write 'tomorrow week' or 'next week' in place of 'tomorrow'.
As the previous letter indicates, Boole had been in London in the last few days.
This letter is addressed from Lincoln.
6 M.H. Gill was the printer in charge of the University Press in Dublin at this
time.
7 See Letter 47.
8 See Letter 41.
This spelling of Shakespeare's name is quite common in the nineteenth century.
De Morgan also omitted the first e (De Morgan 1864c, 201).
Part of the lower fold of this letter has been cut away: hence the omission in
the transcript is unfillable.
This letter is written on black-edged notepaper indicative of mourning. The
notepaper is larger than that Boole habitually used. Boole's mother died on
18 August, 1854.
I have not identified Stevens. This puzzle is explained in Letter 58.
The quotation 'off with the old loves - before he is on with the new' is part
of a verse from a ballad; in Songs of England and Scotland, 1835, the verse
reads:
It is good to be merry and wise
It is good to be honest and true
But it is best to be off with the old love
Before you are on with the new.
The last two lines appear to have become well known in mid-nineteenth century
England; they are quoted by Scott, Dickens, and Trollope.
Cagnoli, Delambre, Puissant, Legendre, and T.S. Davies all wrote works on
trigonometry or mathematical astronomy in which De Morgan's puzzle might
have been expected to appear. He published the puzzle - as a mnemonic - in a
note appended to 18576, 269-70.
The paper which Boole was working on seems likely to have been one of two
major papers he published in 1857: Boole (18576) or Boole (1857a). Boole was
awarded the Keith prize for Boole (18576).
5
BOOKS OLD AND NEW; AND HOMOEOPATHY:
MARCH 1859-MAY 1861
No letters written in 1857 or 1858 appear to have survived; thus the first letter
in this chapter, of 21 March 1859, is over three years subsequent to the last
letter of Chapter 4.
The letters of this chapter contain a good deal on matters of general interest
but relatively little on mathematics or logic. Nevertheless, the years 1857 to
1860 were very productive ones for Boole, who published two long papers one
applying probability theory to the combination of testimonies or judgements
(Boole 18576), the other on the comparison of transcendents with applications
to definite integrals (Boole 1857a). The first of these papers resulted in Boole
being awarded the Keith medal by the Royal Society of Edinburgh (Boole
18576). In addition Boole must have been working on his books on differential
equations (Boole 1859) and finite differences (Boole 1860). In Letter 60
21 March 1859) Boole gives De Morgan a correction in the differential equation
book; in Letter 63 De Morgan acknowledging a copy of the finite difference
book says: 'the book is capital in itself, capitaller as a successor to your
Differential Equations.' Another book, of which Boole says in Letter 60 it 'was
announced for me. The announcement was premature', was never completed.
There are manuscripts in the library of the Royal Society which appear to be the
work referred to in this letter. For an account of some aspects of these manu-
scripts see Hesse 1952.
Some of De Morgan's most important publications on logic appeared in the
years 1857-60. On the Syllogism III and IV (DeMorgan 1858a and 18606) belong
to this period. The fourth paper of the On the Syllogism series was described by
C.S. Peirce as 'a brilliant and astonishing illumination of every corner and every
vista of logic' (Pierce 1931, vol.1, 301). These years also see the publication of
two works which gave summaries of De Morgan's view of logic: Syllabus of a
Proposed System of Logic (De Morgan 1 860a), and the contribution Logic to the
English Cyclopaedia (the volume containing this was published in July 1860).
Letters 60-65
These letters mention the illnesses suffered by Mrs Boole and De Morgan as well
as the birth of Boole's third child. There are frequent references to the books
LETTERS 60-65 75
that Boole and De Morgan had been reading and, in the case of De Morgan,
reviewing.
In the first letter of this chapter, Letter 60 (21 March 1859) Boole says: 'I
have not gone to the diggings yet.' Taken at its face value, this paragraph might
suggest that Boole was still considering leaving Cork for Australia; however, I
think that Boole is here indulging in a piece of jocularity.
One book mentioned in Letter 60 (21 March 1859) was Hamilton's Lectures
on Metaphysics and Logic (Hamilton 1859). Hamilton died in 1856 and never
wrote a full account of his teaching on logic. This work, based upon his lectures
and edited by his former pupils H.L. Mansel and J. Veitch, constitutes the most
complete account of Hamilton's logic. Another book mentioned in this letter is
H.L. Mansel's The Limits of Religious Thought (Mansel 1858).
In this letter also Boole asks De Morgan's advice on the value of an early
edition of Euclid's Elements. Billingsley's translation of Elements was the first
published in English and appeared in 1570. The preface by John Dee is a histori-
cally important statement of the usefulness and relevance of mathematics. The
title page, as Boole says, gives this preface as by 'M.I. Dee'; M.I. perhaps stands
for Master Iohn. The book is both rare and sought after, and now sells for more
than £3000. De Morgan, a bibliophile, owned a copy which is now in the library
of the University of London.
In Letter 61 (9 June 1859) Boole mentions that 'Mrs Boole has been very
ill. . . but is better'. The Booles' third child, Alice, was born in June 1860; in
Letter 65 (17 July 1860) Boole reports that Alice 'is to be made a Christian of
next Sunday'. According to Sir Geoffrey Taylor, a grandson of Boole, 'Alice, the
third daughter, like her father, began, without any mathematical training, to
make mathematical discoveries.' (Taylor 1964a, 51).
In 1860 De Morgan suffered an attack of pleurisy: 'I have had the honour
of a mortal illness for the first time in my life — which few people of the age
of 54 can say.' In Letter 64 (13 July 1860) he tells Boole of his recovery
following homoeopathic treatment. Boole's reply, Letter 65 (17 July 1860)
indicates some scepticism about homoeopathy in contrast to De Morgan's faith
in it: 'if. . . homoeopathy does not produce decided effects soon, do not
sacrifice your life to an opinion. . . but call in some accredited priest of
Esculapius'.
De Morgan's dcpWuos paper, mentioned in Letter 62 (15 September 1859) is
philological rather than mathematical in character. De Morgan distinguished
three senses of this Greek word usually translated simply as 'number'. There is
the general sense of many — in his words 'the notion of many prior to
enumeration' — as well as the familiar concepts of cardinal and ordinal number.
He then discusses the questions: which of these senses was the original one; and
'what is the idea presented to the Greek mind throughout the best period of
philosophical writing?'. He concludes that in Aristotle's writing 'dpifljuoc hovers
between the senses I [ordinal] and II [cardinal] '.
76 BOOKS OLD AND NEW; AND HOMOEOPATHY: MARCH 1859-MAY 1861
In Letter 63 (10 June 1860) De Morgan writes some formulae on finite
differences, it might aid the reader to recall that
Atf* = U x+1 -U x
and that A n m denotes A n x m evaluated at x = 0. De Morgan says that 'Herschel,
I think, calculated [A n m = «(A n_1 m - 1 + A n m_1 )] by real dif-
ferencing. . .'. Herschel wrote a text on finite differences (Herschel 1820) and
this is probably what De Morgan has in mind here.
In Letter 64 (13 July 1860) De Morgan mentions a number of books appar-
ently in answer to a request by Boole ; the titles suggest that Boole was asking for
books on the demonstrative aspect of mathematics suitable for the instruction of
teachers. Most of the books ellipticaUy referred to can be identified with some
certainty, and these are included in the bibliography.
Two remain uncertain: 'Barrow' may be Isaac Barrow — his Mathematical
Works, edited by W. Whewell, was published in 1860; 'Kelland' may refer to
P. Kelland who wrote Lectures on the principles of demonstrative mathematics,
Edinburgh 1843.
Letter 64 also contains some remarks which indicate the opening of the
controversy that De Morgan carried on with Mansel and other successors of
Hamilton. This controversy receives a good deal of attention in the letters of the
next chapter. In this letter De Morgan says: 'I have fired some more shot in the
July number of the English Cyclopaedia 'Logic'.' And later he says in the same
paragraph, 'I cannot imagine what keeps Mansel so long about the logic, unless it
be that he finds very serious difficulties about the novel parts. Hamilton left
them very rough, and he has to defend as well as explain them.'
The English Cyclopaedia was published by Charles Knight, between 1854 and
1862. It was a revised and augmented version of The Penny Cyclopaedia, which
was originally published in parts, then in volume form (27 volumes, plus three sup-
plemental volumes) between 1832 and 1848; De Morgan is said to have been re-
sponsible for about one-sixth of the articles of this work, which was one of the
publications of the Society for the Diffusion of Useful Knowledge. A shortened
form of De Morgan's article on Logic in The English Cyclopaedia appears in
De Morgan 1966, 247-70.
Letter 64 contains one of the few mathematical or logical topics mentioned in
this chaper. The logical notation used by De Morgan in this letter differs from that
which he used earlier (e.g. in letter 12). Now he uses the notation of On the
Syllogism //and writes X) or (X to indicate that X enters universally, while X( or
^indicates that ^Tenters particularly. Thus in a formula he builds up one observes
an even number of brackets; in addition he uses a dot to denote negation (as well
as the previous convention where x denotes the negation of the attribute denoted
by X). Examples taken from On the Syllogism II (De Morgan 1966, 31) are:
X))Y all X are Y
X(-(Y some X are not Y.
LETTERS 60-65 77
Note that Y((X also means all X are Y. De Morgan combines two such
statements:
X))Y{-(Z
i.e. X))Y and Y('(Z. He is then able to derive the inference of a syllogism by
deleting the middle term (letter) and its accompanying brackets and (possibly)
dots. Thus the inference from X))Y(*(Z is X)«(Z or every X is not Z.
60. BOOLE TO DE MORGAN, 21 MARCH 1859
I have not gone to the diggings yet. If any one shall tell you that I have
believe him not. I hereby enter into a solemn engagement not to transport
myself thither without consulting you on the subject. How could you
imagine that I should expatriate myself without at least bidding you good
bye?
You refer to the book on Logic which I have announced or which was
announced for me. The announcement was premature. I have written at
different times as much as would make two or three books but when
returning to a subject I can seldom make much use of old materials. They
have lost their freshness & I can only begin again de novo. And that is
what I am now doing — but — with a modest plan before me, having
certain things to say & only desiring to say them. I am not going to set
aside anything in the Laws of Thought — but only to interpret within the
province of pure Logic what is done there. When this is done I shall quit
the subject for ever.
An edition of Euclid in small folio (not absolutely small but about two
thirds the size of the old folios of the fathers) was shown to me today. It
was the property of a widow who wished to sell it & I was asked my
opinion of its value. It is Billingsley's translation (1570) with a very fruitful
preface by M.I. Dee. No doubt you know all about it. The work is
complete, with pictorial frontispiece etc., but the old binding gone &
replaced by a shabby one of the last century I suppose. I presume the
work not to be of much value — but think it as well to ask you — as I am
writing already.
If you should be looking into my book on Differential Equations
[B 1859] I would wish to tell you that there is an oversight on p.347
where I say (lines 2—5) that it suffices to get n — 1 integrals such as will
suffice to determine p\ , p 2 , . . . , p n in terms of the original variables etc.
It is not proved that any such system of n — 1 integrals will do — though
every such system which rules dx is included in the integrals of the auxiliary
differential equation derived from (45). I intend to add an appendix in
which I shall supply an omission in another part of the work and correct
this error. I now refer to it because I should like to get the best form of
the conditions which ought to be fulfilled — and I am not inclined to think
that Cauchy has got the best. I have an impression that I intended after
writing the passage to reexamine the subject — but forgot it till too late.
I have not seen Hamilton's Metaphysics [1859] but I have read
Mansel's Bampton lectures [1858] which are closely connected with
Hamilton's views & also a review of Mansel in the National [Review]
which I think full of genius. 1
78 BOOKS OLD AND NEW; AND HOMOEOPATHY: MARCH 1859-MAY 1861
61. BOOLE TO DE MORGAN, 9 JUNE 1859
After I had written the enclosed letter to Mr Heaviside I thought on
reflection that I might in the first instance apply to you which I now do
asking you to send it only in case you cannot find a probable answer to
the question yourself.
I hope Mrs De Morgan & your sons and daughters are well. Mrs Boole
has been very ill since her return home but is better. She sends her kind
regards.
62. BOOLE TO DE MORGAN, 15 SEPT. 1859
Your paper reached me safely but I take it very ill that you did not send
me the one on the word aptdixos [D 18596] . I hope if you have not got
one left you will beg one from some one of the people who will not read
and send it to me who will.
The paper you have sent I have as yet only looked into but intend to
read it through.
63. DE MORGAN TO BOOLE, 10 JUNE 1860
I never got your book [B 1 860] till yesterday, for McMillan sent it to my
old address, whence it was returned to him. I am very much obliged to
you: the book is capital in itself, capitaller as a successor of your
Differential Equations. This I say at once. I hope the Cambridge writers
will study these models a little.
By the way (p. 19, 20) I did not calculate A"0 m from the troublesome
(5) of yours [B 1860], but from
A n m = n(A n_1 m_1 + A"0 m_1 )
(p. 255) [D 1842a]. Or rather, Herschel, I think, calculated, by real
differencing, I suspect, and I verified. But I calculated
A"0 m A^O" 1 " 1 A"0 m-1
from + n
2.3... n 2.3... n — 1 2.3... n
I forget whether I discovered these theorems, and saw them afterwards, in
some other book, or whether I got them from another book. I could not
find them again when I looked for them in various likely places.
Poor Bertrand was charged in some foreign journal (Libri told me) with
pillaging me. Now it so happened that he had given every possible proof of
fairness. Independently of his putting my name very prominently forward,
he gave the correct date of my publication, which he had to take, not from
the title page of the book, but by looking at the list I gave of the dates of
publication of the numbers, and dividing the pages by 32. He might have
missed this refinement without any suspicion of fraud. And he sent me his
paper immediately. Not to speak of his method bearing the impress of
another view as clearly as it could do. So it seems that a man cannot
escape, let him do what he will. 3
I hope Mrs Boole and the children are well. With my wife's kind
remembrances.
LETTERS 60-65 79
64. DE MORGAN TO BOOLE, 13 JULY 1860
I have been busy convalescing after an attack of illness. I have had the
honour of a mortal disorder for the first time in my life - which few
people of the age of 54 can say. The name of the beast was pleurisy: but it
did not even put me in danger, though of course a medical looks grave at
danger of danger, and even at (danger) 3 . My symptoms were cut up root
and branch by infinitesimal doses in three days — not rising to a maximum
and then diminishing during the application of remedies: but having their
maximum at the moment of first application. And this I have observed to
be the character of homoeopathic medicines whenever they are to succeed
at all.
I agree with you that the explanation of (*) and ) ( is not down at the
bottom. Quantification is an incident - not the fundamental basis of
explanation.
You must remember that all the rules of validity in p. 19 are one. 4 The
contained of the contained is contained. Thus
X('(Y(-)Z
is X()y))Z
or part of X in y
All y inZ
that part of X in — in Z [ * ]
orinZ [*]
[Note that when he expresses this symbolism in words De Morgan uses the
language of sets. Thus X('(Y(')Z is read
part of X is not in Y and part of Y is not part of Z,
This is the same as X( )y ))Z or
part of X is in not-F and all not-Y is in Z,
hence X{ )Z, i.e. part of X is in Z.
The lines marked [*] are transcribed as De Morgan wrote them; their
meaning is not very clear. The explanation given above contains the gist of
what I think De Morgan intended.]
The splitting and straggling is only the application of this to the varieties
of entry of contraries etc.
I have fired some more shot in the July number of the English
Cyclopaedia 'Logic' [D 18616] . I agree with you about Hamilton. He is a
monster of capability; a monster because so unequally balanced that some
parts are of gigantic development and others only rudimentary. I cannot
imagine what keeps Mansel so long about the logic, unless it be that he
finds very serious difficulties about the novel parts. Hamilton left then
very rough, and he has to defend as well as to explain.
I should think Maynard likely to have Barrow. 5 As you mention
Barrow and Kelland together, I should think that in the same list might
80 BOOKS OLD AND NEW; AND HOMOEOPATHY: MARCH 1859-MAY 1 861
come Gregory (O.) hints to teachers, Young's lectures (Belfast) Beddoes on
Demonstrative Evidence (not much worth) Newman's Geometry — Walker's
Philosophy of Arithmetic — H. Wedgewood on Geometry — and many
others. I have no list of such books.
Our kind regards to Mrs Boole — children well I hope.
65. BOOLE TO DE MORGAN, 17 JULY 1860
I am sincerely glad that you have so completely and so satisfactorily (as to
the manner) recovered from an illness which if not of a mortal character is
at least of a very dangerous one. I have witnessed pleurisy and its former
mode of treatment more than once in my father. One would say before-
hand that homoeopathy could have no effect on such a disease. I remember
hearing of another form of inflammation some years ago treated by
homoeopathy unsuccessfully, and when the patient was in extremity
by the vigorous measures of ordinary practice. This was in London — the
patient a literary man — my informant a clergyman in Lincolnshire who
went up to see his friend, found him getting no better but worse and
insisted on the lancet. My wife's father died of an inflammation of the
stomach under homoeopathic treatment. The moral is — if you are ever
attacked with inflammation and homoeopathy does not produce decided
effects soon, do not sacrifice your life to an opinion, or to the opinion of
any one else, or to a notion of going through with a thing when you have
once begun with it but call in some accredited priest of Esculapius with all
his weapons of war and do as your ancestors did — submit to be killed or
cured according to rule.
I have not seen and fear that I shall not see here the English Cyclopaedia
except the biographical portion — which did not impress me very favour-
ably. The publication of lives of living men is a bad feature I think. If ever
I do get access to the work I will not fail to turn first to your paper.
I am inclined to think that I sent you an Appendix to my Differential
Equations containing corrections chiefly supplied by Mr Todhunter. If not
send me a line at any convenient time and I will forward it to you by post.
Mrs Boole and her children (three) the youngest of whom is to be made
a Christian of next Sunday are quite well. The little neophyte will then be
about three weeks and a half old.
Letters 66-9
Apart from a brief query of Boole concerning Jacobi's work on maxima and
minima, these letters are devoid of mathematics. There are interesting comments
on contemporary literary and military affairs. Also Letter 69 indicates Boole's
friendly relationship with one of his pupils.
In Letter 66 (18 Oct. 1860) Boole points out 'a most audacious instance of
misquotation' in Boase's The Philosophy of Nature (Boase 1860). Boole, claimed
Boase, 'had not clearly apprehended and formulated the principles with which
he commences'. De Morgan's review appeared in the Athenaeum for 18 August
1860 (vol. 33, 222— 3). Boole writes with some heat to De Morgan: 'it ought not
to be passed over without public notice. If you review Dr Boase's book in the
LETTERS 66-9 81
Athenaeum. . . the lash ought to be yours'; he concludes by imputing 'gross
ignorance' to Boase.
In Letter 66 Boole refers to De Morgan's contributions to Notes and Queries.
Some of these were reprinted in the Mathematical Gazette under the heading
Some Incidental Writings by De Morgan: see vol. 9, 78-83, 1 14-22, 126-278;
vol. 10,69-74, 146-9; vol. 11, 157-63, 200-203.
In Letter 67 (13 November 1860) Boole asks De Morgan whether 'you have
particularly examined Jacobi's theory of the criteria of max. and min. in the
Calculus of Variations?'
In Jacobi 1837 were given for the first time sufficient conditions for an
extremal curve to be a maximum or minimum. Previous criteria — due primarily
to Euler and Legendre - were necessary conditions only. This paper had stated
results without proof, so Boole's attempt to discover whether an essay presented
to the Academie des Sciences had been printed shows his interest in the details
of the work. Boole had been interested in the calculus of variations from the
beginning of his research career; one of his earliest published papers was on the
subject, viz. Boole 1840Z».
In Letter 67 there is mention of the book Essays and Reviews 'which was to
raise the greatest religious storm of the century' (Faber 1958, 245). The idea of
publishing the collection is due to H.B. Wilson; it appeared in February 1860.
The book contained seven essays which 'covered almost the whole ground of the
then existing controversies between Anglican churchmen, and between religion
and science' (Faber 1958, 233-4). The most distinguished of the contributors
was Benjamin Jowett. Two of the contributors, Williams and Wilson, were
condemned for denying the inspiration of Holy Scripture in the (ecclesiastical)
Court of Arches - but this was reversed on appeal to the Privy Council. An
attempt to arraign Jowett before the Vice-Chancellor at Oxford failed.
Boole considered Jowett 's essay - titled The Interpretation of Scripture - the
best, while the mathematician Powell's essay - titled On the study and the
Evidences of Christianity - he considered the worst. De Morgan's review of this
work appeared in The Athenaeum, 27 October 1860, vol. 33, 546-9. Also in
Letter 67 Boole paraphrases some remarks that Thomas Carlyle made in a letter
of 4 June 1835 to John Sterling:
. . . assure yourself that I am neither Pagan nor Turk, nor circumcised Jew; but
an unfortunate Christian individual resident at Chelsea in this year of grace,
neither Pantheist nor Pot-theist, nor any Theist or 1st whatsoever, having the
most decided contempt for all such manner of system-builders or sect-
founders - as far as contempt may be compatible with so mild a nature - feeling
well beforehand (taught by long experience) that all such are and ever must be
wrong.
This letter was published in Froude 1884. The substance of the remark must
have been known by 1860, but I have not found any earlier source for the letter.
In Letter 69 (27 May 1861) Boole acknowledges the receipt of De Morgan's
82 BOOKS OLD AND NEW; AND HOMOEOPATHY: MARCH 1 859-MAY 1 86 1
Very learned paper on Tables', presumably his article on mathematical Tables
which appeared in the English Cyclopaedia (De Morgan 18616).
De Morgan's paper is basically a historical and bibliographical report on
Tables. He makes a partial disclaimer, however: 'The list which we mean to give
does not profess to be a bibliography of tables, but will nevertheless give infor-
mation on the subject to all who are not particularly given to mathematical
bibliography' (De Morgan 1861c, col. 978).
De Morgan begins with some remarks about modes of arrangement and
typography to secure a high level of legibility. The main part of the article con-
sists of eight sections in which he discusses tables of the following kinds:
1. multiplication; 2. division and prime number; 3. squares, cubes, and other
powers and roots; 4. pure decimal tables; 5. pure trigonometrical; 6. logarithms;
7. higher mathematical; 8. commercial. Within each section the discussion is
chronological. In his conclusion De Morgan remarks: 'In the present article we
have given about 457 tables, of which 332 are from actual inspection' (De Morgan
1861c, col. 1015). One wonders to what lengths he would have gone had he
professed to write a bibliography!
Euclid's Elements is mentioned again in letter 69 (27 May 1861) where Boole
refers to Peyrard's Greek and Latin edition of Euclid. Peyrard's was the best
edition of Euclid's Elements at this time. The Amj/xcwa are postulates, the
koLvocl ewouxq are the axioms (literally: common notions). In most nineteenth
century editions the axioms included '(a) all right angles are equal', and '(0) that
if a straight line falling on two straight lines make the interior angles on the same
side less than two right angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angles less than the two right angles' (i.e.
Euclid's parallel axiom). However, in certain manuscripts of the Elements these
statements are given among the postulates (usually numbered (4) and (5)). The
latter is now regarded as the better place for these assumptions.
Letter 69 contains a reference to one of Boole's pupils R.A. Jamieson, who
'has been a very distinguished student here. . . also a great favorite of
Mrs Boole's.' Jamieson was an applicant for an interpretership post in the
Far East; he obtained this post, and made his career in the Far East.
After Boole died Jamieson, writing from Shanghai, recorded his impressions
of Boole as a teacher: 'The secret of his success, I think, with senior classes, and
to a limited extent with the junior, was that he never seemed to be repeating or
reproducing what he had himself once learned - he always appeared to be dis-
covering the results he educed, and his students were generally carried along with
him and, as it were, shared in the honour of the discovery.' (Rhees 1955, 76).
66. BOOLE TO DE MORGAN, 18 OCT. 1860
Thank you for the Notes & Queries. I have read all your contributions.
I send what appeared to me at first a most audacious instance of
LETTERS 66-9 83
misquotation but what upon reflection I think may be more charitably inter-
preted. Whatever the cause of the misquotation may be the instance is re-
markable that in the interests of literary honesty it ought not I think to be
passed over without a public notice. If you reviewed Dr [H.S.] Boase's book
[ 1 860] in the Athenaeum it occurs to me that the lash should be yours &
therefore I send the example. It is not the only one but it is the worst.
The charitable interpretation is that Dr Boase read the passage in my
book without understanding it, made a wrong guess at the meaning then
altered the passage so as to give it that meaning & made use of it in order
to display his critical powers — but that having altered the passage so as to
have destroyed almost all verbal resemblance to the original he should have
printed it without quotation marks as if it were a bona fide extract is a
thing not easy to account for charitably in any supposition. The gross
ignorance which it implies is what most men would shrink from the
imputation of even more than from that of wilful falsehood.
I hope you are all well.
67. BOOLE TO DE MORGAN, 13 NOV. 1860
I think you are the author of an article on Essays & Reviews which
appeared in the Athenaeum a week or two ago, & which I read last night.
I judge from internal evidence only. Now after saying that I agree with you
in nearly all your observations. I want to ask you what you mean by
describing the last Essay in the volume as 'sabbatical'. You are too much a
lover of truth to do this merely to complete a parallel or to make a climax
& I have no doubt you attach a meaning & one you can justify to the term
as applied. If you are not busy I wish you would tell me what that meaning
is. I ask you because Jowett's Essay, the one referred to, was to me by far
the most interesting in the volume. I think it the best of the essays & for
the sake of mathematics I am sorry to add Baden Powell's nearly the worst.
Have you ever particularly examined Jacobi's theory of the criteria of
max. & min. in the Calculus of Variations? [1837] Do you know whether
an Essay on this subject which was crowned (I don't know the meaning of
that) some years ago by the French Academy was ever printed? 6
Having written this it occurs to me though I did not think of it before,
that the transition from the Church of England to the Calculus of
Variations is not a very violent one. I suppose the formal difference to be
that in the variations of opinion in matters of theology there is no absolute
max or min though there may be a relative one to the individual i.e. a man
may advance to a certain degree in opinions of one particular kind & thus
gradually recede. If you tell me that Pantheism is an absolute maximum in
one direction of thought I must remind you that Mr Carlyle has discovered
a region beyond these, 'Pantheism Sir! What matters if it were pot theism
so long as it is true?'
I know that you are a hard worker & I feel almost ashamed of sending
this. Don't answer it if you feel it would be trouble.
68. BOOLE TO DE MORGAN, 7 FEB. 1861
I have had most of the papers you sent me arranged and bound — but I
cannot find among them the one you refer to - so if you will be so good
84 BOOKS OLD AND NEW; AND HOMOEOPATHY: MARCH 1859-MAY 1861
as to send it to me I shall be much obliged and if you have no second copy
I will return it. I have no means of referring to Cfambridge] P[hilosophical]
S[ociety] Transactions here after the 8th volume.
69. BOOLE TO DE MORGAN, 27 MAY 1861
A young friend of mine is going up as the nominee from our college for
one of the Chinese & Japanese interpreterships. He called last night and
said that it would be a great pleasure to him if he could get to see
Prof. De Morgan while in London. I asked him why & he replied Why Sir I
have read a great deal of his Formal Logic & also of his Calculus & I should
like to see him. I told him where you were likely to be found (Gower St)
& promised to write to you. I hope you will not take it ill that you have
to bear one of the penalties of fame. My young friend's name is Jamieson.
He has been a very distinguished student here & I am personally very fond
of him. He is also a great favorite of Mrs Boole's. I hope therefore for her
sake as well as mine you will be gracious to him.
And you may tell him for me whether in all the Greek texts of Euclid
as in Peyrard's [1814] the properly geometrical axioms of the editions in
use fall among the AlTrjuocTa & not the Kowai evvoiaq.
I have to thank you for your very learned paper on Tables [D 18616] , a
great deal of which I have read.
Notes
1 The review is in the National Review, March 1 859, 209-27.
2 In the absence of 'the enclosed letter' it is impossible to know what the
question was about. However, the emphasis given to the word probable suggests
that it concerned probabilities.
3 De Morgan gave another - and clearer - account of the accusation against
Bertrand in a letter to Sir John Herschel of 9 August 1862 (see S.E. De Morgan,
312).
4 De Morgan is, I think, here referring to 1 860.
5 Maynard was a second-hand mathematical bookseller (see Graves 1882, vol. 3,
313).
6 An essay was said to be 'crowned' by the Academie des Sciences when it was
awarded a prize medal. It is not clear whether Boole here means that the essay
was written by Jacobi or by another; I think probably the latter.
6
THE CONTROVERSY WITH
HAMILTON'S SUCCESSORS; AND THE JEWS:
OCT. 1861 -NOV. 1862
In Chapter 6 De Morgan's letters are primarily concerned with his controversy
with the successors of Hamilton, and with the fifth paper On the Syllogism (De
Morgan 1863a). The central issue of the controversy relates to the quantification
of the predicate, and I shall begin by giving as much background to the logical
ideas and to the controversy as shall be necessary to understand the letters that
follow. For a resume of the controversy see P. Heath's introduction to De
Morgan 1966.
In his lectures Hamilton had introduced the idea of the quantification of the
predicate into syllogistic reasoning. In addition to syllogisms of the form 'all Ps
and Qs' and 'some Ps are Qs', Hamilton introduced statements such as 'some Ps
are some Qs' and 'all Ps are some Qs'. The background in logic to this idea of
quantification of the predicate is discussed in Prior 1955, 146-56; Prior
described Hamilton as advocating the quantification of the predicate Svith a
quite fantastic incompetence' (Prior 1955, 148).
The controversy arose from a correspondence between De Morgan and
Hamilton in 1846, which led Hamilton to accuse De Morgan of plagiarism. This
seems to have been based on a misunderstanding, in which Hamilton confused
De Morgan's idea of numerically definite syllogisms with syllogisms containing a
quantified predicate; Hamilton made a partial retractment of his accusation, but
it is generally agreed that De Morgan had the better of the argument. 1
In Letter 11, of 1847, De Morgan had remarked to Boole that 'I felt quite
sure he [Hamilton] could not look at logic in any way that could give a view to a
mathematician'. Nevertheless De Morgan maintained an interest in Hamilton's
logic over the next fifteen years. Hamilton had published no definitive account
of his logic when he died in May 1856, his views having become known through
his lectures, and from books published by his former pupils. In Letter 64, of
1860, De Morgan had addressed Hamilton thus: 'He is a monster of capability
because so unequally balanced that some parts are of gigantic development and
others only rudimentary'.
The later phase of the controversy began in 1861 when De Morgan made four
contributions to The Athenaeum entitled Hamiltonian Logic. These appeared in
86 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
July, August, November, and December, on pages 51, 222, 582 and 883-4 of
volume 34. At this time he was in the process of writing On the Syllogism V
which was read on 4 May 1862. This paper gives a comprehensive analysis of
Hamilton's logical ideas and was characterized by C.S. Peirce as 'final and
unanswerable' (Peirce 1931, Vol. 2, 324).
Letters 70-74
These letters are predominantly concerned with the controversy and particularly
with De Morgan's paper On the Syllogism V (De Morgan 1863a). In Letters 70
(16 October 1861) and 74 (1 February 1862) De Morgan rehearses (but not
briefly) ideas of his paper. Letters 71-3 are replies by Boole which, among other
matters, contain his reactions to De Morgan's ideas.
Letter 70 is one of the longest letters in the correspondence and is full of
meat. De Morgan says, in Letter 70, 'In this same number five now [16 October
1861] fermenting, I have some points that can be shortly enunciated'. The letter
continues for 10 sides on these points. He begins with some historic remarks
relating to the singular and plural mode of expressing universals (compare 'each
man is an animal' with 'all men are animals'). He continues: 'I hojd that all and
some emerge a posteriori" and justifies this by a postulate: 'A terrri is that which
divides the universe'. This classificatory approach leads to a consideration of
quantification.
Letter 70 shows the extent of De Morgan's familiarity with the older litera-
ture relating to logic. He refers to ten authors from Boethius to Wallis, mainly in
reference to their exposition of Aristotelian logic. In On the Syllogism V he
makes similar remarks, but these gave slight indications of the particular works
of these authors that he had earlier consulted. (De Morgan 1966, 292.) Several
of these authors are little known.
Paulus Venetus (Paulus Nicolletius Venetus, or Paul of Venice) died about
1429. The Cologne Regents (or Masters) refers to a fifteenth or sixteenth century
edition of Aristotle which I have not traced. Jodochus Isenach (Justus Judocus
of Eisenach) was professor of theology and philosophy at Erfurt; he wrote
Summa Totius Logicae, Erfurt, 1501. The last four all wrote works at the end of
the sixteenth century, or in the first two decades of the seventeenth century: J.
Pacius, or G. Pace, 1550-1633; Burgersdicius, or F.P. Burgersdijck, 1590-1635;
B. Keckermann, 1573-1609; Richard Crackanthorpe, 1567-1624. J. Wallis,
1616-1703, was Savilian Professor of Geometry at Oxford, 1649-1703. His
Institutio Logicae, Oxford, 1687, is the work that Morgan obliquely refers to.
At the end of this long and detailed letter there is an item of a very different
kind, De Morgan describes it as 'a bit of correspondence between physiology and
sociology' in De Morgan's Letter 70 (16 October 1861). This exercise in digital
dexterity concludes with a reference to Sir Creswell Creswell; he was the first
judge appointed to the Divorce Court on its establishment in 1858.
LETTERS 70-74 87
A feature of the correspondence of this chapter is the interest shown by both
correspondents in the Jews and in Jewish culture. De Morgan begins with Letter
70 by expressing his thanks to Boole for 'the extract about Jews' (in a letter
that has not survived) and continues with some remarks that show he read
Jewish newspapers. Letter 71 (4 November 1861) indicates that Boole read
religious poetry of the Spanish Jews. Boole makes further references to his
Jewish studies in Letter 72 (21 November 1861). Boole was still troubled by
financial worries at this time. In the same letter we find him asking De Morgan's
advice regarding advertising for private pupils — presumably a project intended
to increase his income.
The length and detailed arguments of Letter 70 made a considered reply by
Boole a matter of some difficulty. Not surprisingly Boole's Letter 72 begins with:
'I don't think I shall be able to write to you in reply to the 'Logic' portion of
your letter for a fortnight or three weeks'. However, De Morgan is an under-
standing correspondent; he begins Letter 74: 'I shall write you no more logic if
you pester yourself with the duty of answering', and proceeds to expound
further ideas from On the Syllogism V. Boole's reply (Letter 75, 2 February
1862) commences: ' I have received your second logical epistle and have put it
aside with its predecessor to be studied in due time'.
In Letter 73 (7 January 1862) Boole gives expressions of support for De
Morgan in the controversy: 'you seem to me to be quite in the right — at any
rate substantially so'.
In Letter 74 (1 February 1862) De Morgan is concerned with Hamilton's use
of the term 'some'. De Morgan distinguishes three senses — which he claims
Hamilton had confused. There is a non-partitive sense; in this sense some indi-
cates only not-none. There is the singly partitive sense in which some indicates
some-not-all, with no assertion about the remainder. And there is the doubly
partitive sense which indicates some-at-most, with the remainder not possessing
the particular attribute (De Morgan 1966, 276-7). This leads to De Morgan's
analysis of Hamilton's 'poor syllogisms'. Seventeen of the 36 are adequate, 4 are
'purely Aristotelian'. But 'There remain 15 . . . and all are vicious'.
70. DE MORGAN TO BOOLE, 16 OCT. 1861
I am much obliged to you for the extract about Jews. They were a well
trounced community. I have no doubt they exhibited a counter feeling
when they dared: and so increased the hatred against them. Shylock must
have been a picture of things which had happened, even if exaggerated.
The Jews now dare to publish their own newspapers, in which they
treat Christianity as blasphemy. I get hold of one now and then. In one
there was a very moderate article, which spoke of the Rabbi Joshua of
Nazareth [Jesus Christ] and the Rabbi Saul of Tarsus as teachers of con-
siderable merit, regretting that their injudicious followers had interlarded
their teaching with mythical biography and forged miracles. My! how the
age advances. Whether the writers, or some of them, in Essays and
Reviews, mean the same thing, is a matter of curious speculation.
THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
Being always grubbing at logic in some state or other, I have been at Sir
W.H. - my best friend, whom I treat with prepense ingratitude - and you
may see by the Athenaeum [vol. 34,51,222] that I convict him of actually
forging invalid syllogisms. His friends are quite silent. Can you do anything
for him? I mean in the way of suggesting a possible meaning for his
quantifications which will make his syllogisms valid. Dr Mansel, who
rushed in so boldly to the defence of his mathematical blunders, and left
me the field at last, was wise enough, when I wrote my first letter in the
Athenaeum to write me a private note, saying that he had intended to
write to the Athenaeum - but thought it was really so simple a thing to
answer that he must have mistaken my meaning. He thought that it was
etc. etc. I gave him a spicilegium 2 of what I should think - and he
abandoned his intention. That I should be allowed to advance, without
contradiction, that the great reformer - who blew his own trumpet so
very loud - has committed actual paralogisms, is so strange, that I really
want an opponent for the sake of the case and the subject. Mr (I mean Dr
- meum cuique) Mansfield Ingleby — has written to say that he cannot
allow me to assume etc. etc. without contradiction. I have answered that
he must publish, if he wants me to notice. He is the man who signalised
himself by a very curious blunder a year or two ago. If you correspond
with anybody who is inclined to contradict me in private, I wish you
would put him up to firing a shot in public. It is nothing at the time that
no answer appears: but if I be allowed to recapitulate in my Cambridge no
V [D. 1863a], which is on the stocks, and to say that there was no reply, it
will have a great deal of meaning 20 years hence. Besides I am personally
in want of a row: as the Irishman said, I am dry-moulded for want of a
bating.
In this same number five, now fermenting, I have some points which
can be shortly enunciated. Two of them are as follows.
1. On coming to examine Aristotle etc. independently, I find that all the
old logicians from A. downwards are singular, monadic, exemplar, - as
you please - in their enunciations. Their universal is always each one,
omnis, naq, - singular: their particular is always some one, Quidam,
aliquis, tic. When I made Hamilton's system exemplar, in 1850, 1 had not
thought of this in the comparatively little enunciation I had made then.
The logicians now all read plurally or rather collectively - by lumping the
individuals into the extent of a term. So that with them Omnis homo est
animal means all (the extent of the term) man is in (the extent of the
term) animal. But the ancients meant no such thing - by omnis homo
they mean every man. At this moment I remember Aristotle, Boethius,
Paulus Venetus, the Cologne Regents, Isenach, Pacius, Burgersdicius,
Keckermann, Crackanthorpe, who are all unmistakeable. The pluralisers
are such moderns as Wallis, the Port Royal, etc. - and most of them are
rather mathematical. If you want the latest collection of the purest
Aristotelian, Crackanthorpe is the man. He collects all the quantifiers; 12
in number, all singular! So I believe that I hit upon the extension of the
old systems when I cut down Hamilton's plurals into singulars. What do
you make of Hamilton's form for (B) 'No X is 7' i.e. Any X is not any Y
(A).
He affirms that 'any' is exclusively limited to negation: I say any one
knows better than that. But I want to know whether, in English (A) and
LETTERS 70-74 89
(B) are of identical meaning. Speaking of English merely, of course I see
that any man is not any stone. But should I not be obliged to say this of
stone man if there were one. This man is not any stone: there is one bit of
stone he is, and no other. That 'no fish is a fish' is rendered by Hamilton
'Any fish is not any fish'. But to my idea this seems true. Any fish is not
any fish - any fish is but itself and not any other fish. Turbot is not
salmon. There is a prime ambiguity, it strikes one, about the meaning of
any: is-not-any fish and is-not any-fish mean to differ in meaning.
2. I have been overhauling the sources of enunciation, and I conclude that
the primary introduction of quantification is an error and a misfortune: I
hold that all and some emerge a posteriori. I have touched this point in my
third paper [D 1858c], but not to the extent of showing the basis of my
eight forms independently of explicit introduction of contrary terms. I
now get it as follows. I prefix all necessary matters.
Postulate. The universe — any assigned extent of thinking ground.
Postulate. A term is that which divides the universe. A name which
embraces the universe cannot be the means of affirming or denying — that
is of dSff that whi ch might have been ^eS^i in that universe. Hence
the lowest term is singular, an individual: its contrary, the highest term, is
penultimate, containing all but an individual.
Postulate. Any individuals whatever in the universe, which are not all, may
be the contents of a term. The name may be wanting in language, but the
power of separating these individuals exists in thought. And one need not
look for a differentia for there must have been one before or in separation.
For instance, I separated as a species of 'material object' - 'All the men
who have killed their brothers; the hundred largest inkstands that ever
were made; and the first Gaul who set eyes on Caesar'. What is the
differentia of this species? — It is 'selected by the fancy of a logical radical
to illustrate the unlimited power of division of the universe'. The old
logicians would say that these objects are not a species because the
differentia is not of their essence: but I defy them to show this. If all
things be predestined from all eternity — it was as much of the oirna. of the
inkstands that they should be thus associated with the fratricides as that
they should be capable of holding ink: there are no degrees of necessity.
So much for the essence.
Pragmatic enunciation is comparison of terms as to contained or not
contained. It is of two kinds: —
1. In which subject and predicate are compared by difference of relation
to an indefinite third term (Inconvertibles).
2. In which subject and predicate are compared by sameness of relation to
an indefinite third term.
N.B. This indefinite third term avoids the necessity of bringing in other
comparing relations, as excluded etc.
Let X) be X is in do (X
- X{- be X is not in do -)X
Let X( be X takes in do )X
— X)- be X does not take in do (-X
90 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
1. X))Y X is in, Y takes in, - some third term
(N.B. the third term may be X or Y itself)
X((Y X is not in, Y does not take in, any third term (i.e.) not both
X((Y X takes in, Y is some third term.
X))Y X does not take in, or Y is not in, any third term.
2. X)(Y Either X does not take in or Y does not take in any third term.
X()Y X takes in, and Y takes in, some third term.
X()Y Either X is not in, or Y is not in, any third term (i.e. Everything
is either X or Y: the universe is not a term)
X)(Y X is in, and Y is in, some third term.
The affirmations are conjunctive and the third term is particular — some
one. The negatives are disjunctive, and the third term is universal - any
one.
My usual reading of (•) and )•( has very much bothered several who have
spoken to me about it. Here ( •) is as negative as the rest, etc.
Here the spiculae [i.e. (•), ) etc.] are not directly quantitative — and
'all' emerges from 'is in' and 'does not take in' - while 'some' emerges
from 'takes in' and 'is not in'.
The other modes of reading turn out ineffective. For example
X is in, Y takes in, any one third term. Not true of X in any sense, nor of
Y, unless it can be the universe. And so on.
X is not in, and Y is not in some third term. True of every possible pair,
unless the universe have but two individuals, one X and one Y.
And so I could write on through a plusquam legible number of sheets. I
will only add that when one comes to quantify there are two modes of
reading — by part and by whole : a whole of X meaning any thing which
contains X, from X itself upwards. And
Any part requires some whole
Some part requires any whole.
Thus
X))Y
any part of X is in some part of Y
any part of X is in any whole of Y
Some part of X is in some part of Y
Some whole of X is in any whole of Y
etc. etc. etc.
This is the ground work of the change of quantities, in passing from exten-
sion to intension.
But it will take a life to get hold of the wider aspects of onymatic rela-
tions which have given rise to forms of language — But no one form [is]
completely octagonal. It reminds one of the Indo-Germanic philology,
here a form essential to a bit of system is utterly absent in one language,
and turns up in another.
Here is a bit of correspondence between physiology and sociology. The
two hands are to be joined together in the following manner [Fig. 1 ]
LETTERS 70-74
/^ >^K V
^-B
91
Fig. 1.
A meant for thumbs. Father and son
B forefingers Mother & daughter
C second fingers Presently explained
D third Husband & wife
E fourth Brother & sister
It will be found that father & son can separate - do, mother & daughter -
do Brother & sister - but that husband and wife cannot - But let the Cs
rise up and join - This is Sir Creswell Creswell - then husband & wife can
separate.
With our kind regards to Mrs Boole & the etc. etc.
71. BOOLE TO DE MORGAN, 4 NOV. 1861
As you are great in all sorts of Arithmetical antiquities I send you an
extract from a very remarkable book which I have lately read, 'Die
Synagogale Poesie des Mittelalters', Dr [Leopold] Zunz, Berlin 1855. The
book contains two terrible chapters on the sufferings of the Jews. In the
second of these and referring to the seventeenth century occur these
words, p. 345:
'Sogar in den Rechenbuchern wurden die Juden verfolgt, wo die Kinder
die Zinsen ausrechneten, die Joseph der Wucherer einem mitteleidenden
Christen abnahm u.dgl. Es gab damals weder einen Unterricht und ein
Buch, noch ein Gesetz und eine Sitte aus denen nicht, von friihenter
Jugend an, der Judenhass genahrt und so zur zweiten Natur geworden
ware.'
[Translation: Even in books on arithmetic the Jews were persecuted, when
children calculated the rate of interest which Joseph the Usurer charged to
a compassionate Christian, etc. At that time there was no instruction or
book, nor a law or custom which did not, from earliest youth onwards,
nourish the hatred of the Jews, which thus became second nature.]
92 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
72. BOOLE TO DE MORGAN, 21 NOV. 1861
I don't think I shall be able to write to you in reply to the 'Logic' portion
of your letter for a fortnight or three weeks. And I very much fear that
though I am in the land in which a disposition to fight is supposed most to
bloom I shall not be able to find you an antagonist. You could not contrive
to bring in the Pope or Garibaldi or the Queen's Colleges — and without
something of the kind I fear I can do nothing for you here. If I entered the
lists myself it would be on your side and that would only make your case
worse. Shade of Duns Scotus arise!
As to the Jews, the fullest information which I have seen about their
present state & movement is contained in Vols I & X of 'Die Gegenwart' —
a sort of supplement to the Conversations-Lexikon. Vol. X gives all the his-
tory, up to the time of its publication, of movements in Germany for
getting rid of the ceremonial law, which some wish to do wholly, of
renouncing all national distinction, etc. etc. — including even all veneration
for and hope of ultimate return to the Promised Land. A member of this
party in Frankfurt refusing to have his child circumcised, the hadebonary
party actually went the length of calling upon the magistrates to deprive
him of the civil privileges accorded to the other Jews. 3
I fancy from what I have read that the literary portion of the Jews have
been more deeply influenced by modern destructive criticism than the
corresponding class among Christians — perhaps because the Jewish cere-
monial must be felt now as it was in the days of Paul to be a grievous bur-
den too heavy to be borne.
There is a curious fact about the religious poetry of the Jews of the
Middle Ages the quantity of which as you probably know is very great. It
is that the only really good poetry was written by the Jews of Spain
especially under the Mohammedan princes. I noticed this myself in the
book I mentioned compared with another, 'Die religiose Poesie der Juden
in Spanien' [Sachs 1845] — before I saw that it had been observed before.
I believe the cause to have been that in Spain in the 10th, 11th & 12th
centuries the Jews were less oppressed than in other countries and at
subsequent periods. Shelley's lines
Men
Are cradled into poetry by wrong
They learn in suffering what they teach
in song
contains with some truth a great error. 4 Freedom for the development of
faculty is essential to all intellectual products that are of any value. When
sorrow falls upon a nature that has before been cultivated under favour-
able influences it may call out poetry — but a state of continued oppres-
sion and misery dwarfs all the faculties alike. Accordingly the poetry of
the Synagogue of the middle ages — is for the most part one long wail.
This is quite enough about the people of Israel. I now want to ask you
for some advice. I have got my house enlarged and want to take into it a
few pupils. I should greatly prefer to get a few from England. Communi-
cation is now so easy that this ought not be difficult. My plan would
be for my pupils to attend the classes in the College here and for me to
keep a general supervision over all their studies and help them in their
LETTERS 70-74 93
difficulties in most. Now is there any objection to my advertising in some
such form (I have obtained the requisite sanction for receiving pupils) as
thus 'Professor Boole wishes to receive a few pupils into his house. Terms
& conditions may be known on application' — or is there any better way.
The mention of my name would perhaps be my best mode of getting
pupils here — for I fear the vague generality of 'A Professor in a College'
etc. would not do much for me. I have done nothing yet.
With kind regards from Mrs Boole as well as myself to Mrs De Morgan
etc. etc.
P.S. I should feel obliged if you would send me a Jewish newspaper some
time or other — which I would return if you wished.
73. BOOLE TO DE MORGAN, 7 JAN. 1862
I ought to have written before to thank you for your answer to my
question on your criticism of Jowett 5 — the answer is quite satisfactory -
and also to say that in your controversy with the defenders of Hamilton's
reputation you seem to me to be quite in the right — at any rate substan-
tially so. The delay has this good in it - that it gives me the opportunity
of sending you and yours our united good wishes for the coming year — I
should rather say for that part of the new year which is yet to come. Per-
haps I may have the pleasure of seeing you again before long. I think it
not unlikely that Mrs Boole will accompany me on a visit to London at
Easter but nothing is settled yet.
I have nothing to tell you except that very great changes are talked of
a new college — beneficial ones I think in every way. I have had a good
deal of work lately most of it voluntary and not I think and hope quite
without result. With best regards to Mrs De Morgan in which Mrs Boole
joins with me.
74. DE MORGAN TO BOOLE, 1 FEB. 1862
I shall write no more logic if you pester yourself with the duty of
answering. I write as I am moved - by what spirit I cannot say till I have
talked the matter over with Hamilton in the next world. In the meanwhile
I have completed my examination of him in this, and the results are
amusing.
Bear in mind that what he says of some he means to deny of all the
rest. Now first, he has actually forgotten to carry his system through. In
his
Some X is not some Y
he leaves his some singly partitive. If he had carried his system into this
proposition, queerly enough, his 'Some X is not some V would have been
the equivalent of the old Aristotelian 'Some X is some V - the simple
denial of 'No X is Y\ As follows
Some X is not some Y =
Some Y is not some X (by hyp.)
Therefore all other X is Y
all other Y is X.
94 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
This accords with
X 1 1 1 1 1 1 1 1 1 1 1 1 X
Y 1,| 1 1 1 1 1 1 1 1 1 n 1 1 1 1 1 1 1 1 1 1 Y 1 1 1 , 1 1 1 1 1 1 1 [ 1 1 1 1
1 1 ii i i 1 1 1 1 1 1 1 1 1
The 1 1 1 1 1 1 being subject and predicate.
Every relation is here except X ___ —
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — ■■ 1 1 1 1 1 1 1 1 1 1 1
Now as to his poor syllogisms. They are 36 in number. 15 are saved by
containing the identiy 'All X is all Y\ No other premise, with this will
make a false syllogism, or a difficult one. For there is nothing to do but to
insert X, equivalent of Y, in the other premise. In these 1 5 no peculiarity
of H's system does work.
2 are saved by having his 'Some X is not some Y > as a premise, and con-
clusion also. But his X{.)Y, as he says 'quadrates with all the rest'. But
these two would be good if X(.) Y had been systematically treated.
4 are purely Aristotelian — H's system does not work any change of
meaning in their premises or conclusion. There remain 15, in which his
system works its will: and all are vicious.
1 1 are absolutely false, and turn men into stones, or anything else you
please.
4 are only incomplete — but would be complete and valid if (.) were
treated by the system.
I have written to Mansel to know what he took Hamilton's system to be
in 1851, when he spoke in such high terms of it. From some of his expres-
sions he clearly acknowledges some departure from the old meaning of
'some'. But whether by single or double partition I cannot make out. He
has not yet answered. If he meant double partitiottj he is in an ugly fix:
but I suspect he really only thought of the single.
I hope all is well with you. My wife unites in kind regard.
Letters 75-82
In these letters the controversy about Hamilton's logic rumbles on. In Letter 77
(20 September 1862) he mentions his intention tt> make a 'last appeal to
Hamilton's successors' in order to find out from thern exactly what Hamilton
meant by 'some'. This appeal was contained in a letter in The Athenaeum, 18
October 1862, which led to a correspondence in the columns of that journal
with Baynes. De Morgan summed up the results of this correspondence in an
Addition to On the Syllogism V, dated 26 December 1862 (De Morgan 1966,
337-45).
De Morgan begins this letter with some unusually astringent remarks' 'I am
much obliged to you for your last Qn probability. Having nothing at all on hand
except a valuation, an introductory lecture, an examination, a paper for the
Cambridge Trans., a last appeal to Hamilton's successors. . . with some odds and
ends not worth recounting — I shall be able to give it a full reading in less than
six months'.
* LETTERS 75-82 95
An item of general interest in this letter concerns one of De Morgan's intro-
ductory lectures at University College London. The lecture was on 'the method
of examining at Cambridge'. This lecture was never printed, however (S.E. De
Morgan 1882 278-9.
Letters 79 (6 November 1862) and 80 (7 November 1862) contain a number
of interesting remarks on logic and other matters. Perhaps the most interesting
remark is made in Letter 79 (6 November 1862) when Boole asks De Morgan for
his thoughts on 'Renan's declaration that to the Semitic mind expressing itself in
the Semitic forms of language no logic was possible'. As well as his well-known
La Vie de Jesus, Renan wrote on linguistics. This remark (which I have not
traced) may have occurred in L'Histoire des Langues Semitiques, 1847 or in
Histoire generate et systeme compare des langues semitiques, 1855. De Morgan's
reply — in Letter 80 (7 November 1862) — is an anecdote on the Persians; a
curious error on De Morgan's part, as the Persians are not Semites.
In Letter 79 Boole remarks: 'I have been studying a bundle of letters of yours
on Logic ... [I] feel I have nothing to say about them but I have been interested
in them very much. But it has been like reviving an interest that had died . . .
However I . . . look forward to the time when I shall study Logic again'. Near the
end of the same letter: 'I will look at the syllogisms of Hamilton but I shall do it
for your sake not his'.
In Letter 70 (16 October 1861) De Morgan had proposed the notion of a
terms as 'that which divides'. Boole was not convinced and expressed his doubts
in Letter 80.
Boole seems still to have felt isolated in Cork. In Letter 79 Boole writes plain-
tively: 'There is absolutely no person in this country except myself with whom
I ever speak on subjects like this [i.e. logic]. I feel this as one of the many draw-
backs in living in this country. . . '. But De Morgan replies trenchantly in Letter
80: 'I have not one person to whom I can speak on logic ... I go from one
month to another without any conversation on my studies with a person whom I
cannot claim to teach . . . therefore I warn you against the notion that you are a
mental Robinson Crusoe'.
But there are also matters unconnected with the Hamilton controversy. In
Letter 78 (4 November 1 862) Boole sends De Morgan a quotation from Leibniz
on logic. The extract is from a letter Leibniz wrote to Wagner and may be found
in Leibniz 1875, Band 7, 514—27. An English translation appears in Loemker
1956, (see third edition 1969, 462-71).
Letter 75 contains a detailed account of some research on differential equa-
tions; near the end he remarks: 'I fear that I may have said enough to tire you.
Pray do not feel any scruple about treating my letters in the way you ask me to
treat yours'. We may feel surprised that people in those times found the time to
indulge in extensive letter writing; that they did shows the value they placed on
correspondence. But the longer letters that passed between De Morgan and
Boole were not easily answered. Boole is concerned with eliminating a number
96 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
of differentials from a system of partial differential equations. As the process is
one of some complexity and Boole's notation is rather cumbersome, I shall give
an account here in present-day style but adhering to Boole's notation as far as is
practicable. Roman numerals are used to distinguish equations I have numbered
from those numbered by Boole.
Boole begins with an equation
n
£ <hj(xi ,x 2 ,...,x n ) dxj = 0, (I)
;' = i
where l</<«— r, and seeks a function P(x u x 2 , . . . ,x n ) such that the
equation
n
I bjPdxj = (II)
/-i
is compatible with (I). Any such P is an integral of (I). Solving (II) for dx„ and
substituting the resulting expression for <bc n in (I) yields:
n-l
E {4HnW-<h*nP}tej = (III)
j = l
where l</<«— r. Using these n— r equations we may eliminate 6x n - u
dx n _ 2 , . . . , dx n _ r+1 finally obtaining the single equation
t */dx y = 0- (IV)
The coefficients \pj, 1 </ < r are linear in bjP. Now set these i// y - = 0; in the case
r=2 these equations yield Boole's equations (1) and (2). From this point,
following Boole, I shall assume r = 2. Thus we now have Boole's equations
AxP = I AjdjP = (1)
j' = i
n
A 2 P = Y. BfrP = 0. (2)
i' = i
Form now (A X A 2 — A 2 A X )P = 0. The second order partial derivatives cancel out
in pairs and we obtain
A 3 P = (A 1 A 2 -A 2 A 1 )P = t KjbjP =0 (V)
where
Kj = £ (AiBji-AjM,
i=l
(here B n means bBj/dxt etc.). Using (1), (2) to eliminate d n P, d n -iP from (V)
we obtain n-2
A 3 ^ = I CjdjP = 0. (3)
y=i
LETTERS 75-82 97
Boole envisages repeating this argument, forming A 2 A 3 — A 3 A 2 and hence A 4 P
etc.. Now it may be that A 3 P is identically zero; in any case the construction of
successive A k P will ultimately become identically zero and the process
terminates. Let A m P be the last non-identically zero expression and consider
A t P = A 2 P = . . . = A m P = 0.
The final part of the argument consists of forming
m
£ Xy V = °>
which may be written j = 1
Xi I AjdjP + \ 2 £ BjbjP+ . . . = 0.
j=i y=i
a differential equation which yields
dxi dx 2 djc„
\ 1 A l + \ 2 B 1 + . • . \iA 2 + \ 2 B 2 +. • • '" Xi^„ +X 2 5 n + . . .
In conclusion Boole envisages 'eliminating the Xs [giving] a system . . . which
will be capable of reduction to exact differentials'.
Two points need to be made in conclusion. First: in Letter 75 Boole made
one or two minor slips writing n + r for n or n for n — r. I have corrected these
slips. Second: Boole had intended including a discussion of the ideas of this
work in the second edition of his Treatise on Differential Equations . The supple-
mentary volume, edited by I. Todhunter (Boole 1856) included related material
in Chapters 25-7.
Letter 76 (21 April 1862) mentions a correspondence carried on at this time
between Boole and Cayley. 'Mr Cayley', writes Boole, 'wrote to me about an old
subject of dispute between us . . .'. The subject was the question on probabilites
which Boole put to De Morgan in Letter 33 (24 July 1851) and which was refer-
red to in Letters 34-7 (July-August 1851). Cayley wrote a paper giving his
answer to the question (Cayley 1853); Boole disputed Cayley's interpretation of
the question in Boole 1854ft. Cayley summed up this phase of the controversy in
a note in volume 2 of his Mathematical Papers (Cayley 1889, vol. 2, 594-5).
The disagreement hinged upon the matter of what should be assumed relating to
the independence or otherwise of events.
In 1862 Cayley again tried to settle this question and sent Boole his new
thoughts; Boole again did not wholly agree with Cayley and wrote him a letter
setting out eight observations. Cayley included these as part of his published
solution (Cayley 1862**). Perhaps the most important outcome of the renewal of
this correspondence was that it led Boole to a general determinantal criterion for
'the conditions of analytical validity of the method . . .', which he published in
the Philosophical Transactions of the Royal Society (Boole 1862a).
Boole's query in Letter 76, 'Are you ever disposed to see Ireland (I have seen
98 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
enough of it) travelling is easy now . . .' indicates the effect the development of
the railways had on travel in Britain in the period of the correspondence. By
1850 the railway route from London to Holyhead and thence by boat to Dublin
was open; also Dublin and Cork were connected by rail by this time. However,
this suggestion must have fallen on barren ground. De Morgan rarely left London
and writing to W.R. Hamilton in 1853 he siad: 'I never got further north than
Cambridge, and never while at Cambridge penetrated to the northern extremity
of the town. So much for me as a sight -seer and traveller. And yet I have been in
three-quarters of the globe — in arms — not as a combatant but as an infant'
(Graves 1882, vol. 3,462).
75 . BOOLE TO DE MORGAN, 12 FEB. 1862
I have received your second logical epistle & have put it aside with its
predecessor to be studied in due time. The subject at which I have been
working is not logical though it contains a good deal of applied Logic & I
was once obliged even to have recourse to my own Calculus of Logic in
order to guide me through the maze. The results connect themselves a
good deal with some of your own speculations & I will give you a brief
account of them as I think they will interest you.
The subject is the theory of the solution of simultaneous differential
equations in which the number of variables exceeds by more than one the
number of equations so that we cannot say beforehand even that an
integral in the ordinary sense of the term exists. I wished to discover a pro-
cess for determining how many integrals such a system admits & how they
may be found i.e. how their determination may be made to depend on the
always theoretically possible solution of differential equations in which
the number of variables exceeds by one the number of equations. I have
succeeded in this and the method is as follows.
1st A system of n — r simultaneous differential equations of first order
and degee between n + r variables can always be reduced to a system of r
partial differential equations linear and of first order (Let P = c be an
integral of the system, eliminate dxidx 2 dx n between the system and
dP = and equate to the coefficients of the other differentials).
In what follows I will suppose r = 2 as the theory of other cases is
involved in this.
2ndly In the supposed case of r = 2 we have two partial differential
equations of the form
dP dP dP
Aif-+A 2 f-... + A n — = (1)
d*! dx 2 dx n
dP dP , dP
B i^- + B 27- ■ • ■ + B »T~ = ° (2)
dx\ dx 2 dx n
If we multiply the second by A, add to it the first, form the auxiliary sys-
tem of common differential equations by Lagrange's method and then
eliminate X we fall back on the original system of differential equations.
Instead of this proceed as follows.
LETTERS 75-82 99
3rd Let
d , d A d
Ai = Ai— +A 2 — . -An—
djc x dx 2 QXn
d , d „ d
A 2 = B l — + B 2 —...B n —
djc! dx 2 dx n
then the partial differential equations are
AjP = 0, A 2 P = 0.
Form now the partial differential equation
AjAjP-AaAiP =
or for simplicity
(A 1 A 2 -A 2 A 1 )P = 0.
dP dP
This will be linear like (1) & (2) and if we eliminate will be
of the form & x n-i dx n
dP , dP dP
C t + C 2 ^- . . . + C n _ 2 = (3)
dxi dx 2 dx n - 2
The equations AjP = A 3 P = may be so prepared that this equation
shall be obtained directly without the final elimination.
Represent (3) by A 3 P = and repeat the process between it & either or
in succession both of the former equations we shall thus get two new
equations
A 4 P = A 5 P =
one of which shall have one, the other two, differential coefficients of P
fewer than A 3 P = 0.
This process must be continued always in the form of diminishing the
number of differential coefficients till it stops i.e. till i^q new partial
differential equations arise.
Suppose in this process m partial differential eqvjatioris have been
obtained then the original system of common differential eqiiaticms will
have just so many integrals less than it has equations as there have been
partial equations obtained i.e. additional to the two by which th§ given
system was replaced.
To find the integrals let
AiP = A 2 P =0 ... A m P =
be the whole system of partial differential equations. Then forming the
equation
A 1 P + X 1 A 2 P... + X OT _ 1 A m P =
and by Lagrange's method its auxiliary system, & lastly eliminating the Xs
we have a system of final equations which will be capable of reduction to
exact differentials & will give the integrals sought.
I will not go into the Logic of this further than to note that from the
100 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
form of the symbolical equation
(A^j -A 2 A l )P = (A)
and from the nature of the symbols A u A 2 any simultaneous solutions of
AiP= A 2 P = will satisfy (A) - secondly that (A) expresses the con-
dition which must be satisfied in order than an integral of the given system
obtained by making a superfluous variable constant may by variation of
parameters become an integral of the system unreduced.
One of the most important applications of the method is to the deter-
mination of the conditions under which
Rr + Ss + Tt + s 2 - rt = V
admits of different forms of integration. Here is one result. The necessary
& sufficient conditions for the above admitting an integral expressed as the
envelope of
z = 0(x, y, a, b, c)
a, b, c being subject to two arbitrary connecting conditions are the follow-
ing viz. S 2 - 4(RT - V)
(1) m 2 — Sm+RT—V= 0. [Crossed out by Boole]
(2) A t R +A 2 m =
(3) A im +A 2 T =
in which m is one of the equal roots of
m 2 ~Sm+RT-V =
and
A t = m — +T — + p—
dx dq dp dz
d , d d , d
A 2 = r R m 1- q —
dy dq dp dz
The first condition was given by Ampere and yourself. I have never seen
the others.
I fear that I may have said enough to tire you. Pray do not feel any scruple
about treating my letters in the way in which you ask me to treat yours.
We are all very well & give our kind united regards to Mrs De Morgan.
P.S. If you happen to know of anything of this kind having been done
before I need not say that I should be obliged by your letting me know.
Please to keep my letter.
76. BOOLE TO DE MORGAN, 21 APRIL 1862
I was very glad to receive your tract this morning because not having
heard from you for some time and also not having recognised your hand in
the Athenaeum we began to think you were ill. I dare say you were busy.
I have been myself hard at work on the Theory of Probabilities. I had
just finished my original on differential equations which I sent you a short
LETTERS 75-82 101
account of - had not and have not yet drawn up a paper on the subject,
when Mr Cayley wrote to me about an old subject of dispute between us
adding however that he was strongly inclined to believe in my theory of
Probabilities developed in the Laws of Thought but could not understand
the metaphysics of it. I do not know whether I have made him do so now
but the correspondence has led me to resume the analytical discussion of
my method which I had vainly attempted to complete before - this time
with success. I have proved that in all cases the conditions of analytical
validity of the method are simply the conditions of consistency in the data
— what I have elsewhere termed the conditions of possible experience.
I do not think I have ever engaged in as difficult a mathematical investi-
gation. The most important part of it consists in proving that a certain
functional determinant is always positive whatever the number, of the vari-
ables n. When I tried to do this before I could not get beyond n = 4 and
the difficulty of getting thus far you may imagine when I mention that
while for n = 2 the number of terms of the determinant is 4 — for n = 3 it
is 52. That is a tolerably rapid rate of progression. For n = 4 it is some
monstrous number which I never took the trouble to investigate.
I was going to get about Logic including your letters when Cayley's
letter came and when I once got fairly started in the inquiry I could not
stop. Now I must finish both papers before taking up the Logic.
We hope you are all well. Are you ever disposed to see Ireland (I have
seen enough of it) travelling is easy now and you might be the better for a
change. If so it would be the greatest pleasure to see you here and Mrs De
Morgan with you if she could come. We are all pretty well.
77. DE MORGAN TO BOOLE, 20 SEPT. 1862
I am much obliged to you for your last on probability. Having nothing
at all on hand except a valuation, an introductory lecture, an examination,
a paper for the Cambridge Trans, a last appeal to Hamilton's successors to
know what his system was, and an approaching session, — with some odds
and ends not worth recounting — I shall be able to give it a full reading in
less than six months.
I am going to try whether by any sarcasm I can get a pupil of Hamilton
to say what he taught as the meaning of 'some'. Spencer Baynes, the pupil,
the prizeman, the substitute, and the accredited expounder will not answer
a letter on the subject — I wrote to him months ago, and he will not even
acknowledge receipt. Mansel does not know: Fraser neither knows nor can
find out. But my letter to the Athenaeum — which I intend for the latter
end of October, will announce that registered copies of the Athenaeum
containing it are to be sent to Mansel, Veitch, Sp. Baynes, and Fraser,
editors, substitute, and successor. And I expect that even then there will
be no answer.
I hope Mrs Boole and the little ones are well.
78. BOOLE TO DE MORGAN, 4 NOV. 1S62
I congratulate you on having got an antagonist at last. I shall look for
the next forthcoming numbers of the Athenaeum with interest and I will
read over also your letters to me which I had taken out of the repository in
102 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
which they were kept, in order to examine them again, just before I saw
the letter of Mr Baynes.
Here are a couple of lines which show what Leibnitz though about the
Logic of the past and that of the future.
— 'so muss ich zwar bekennen dass alle unsere bisherigen Logiken kaum
ein Schatten dessen sein so ich wiinsche und so ich gleichsam von feme
sehe'. Schreiben an Wagner.
[Translation: 'Indeed I must admit that all our logics up to now are a mere
shadow of what I would desire and what I see from afar'. Written to
Wagner.]
As you once mentioned that you are not much in the habit of reading
German I add that so in the German of Leibnitz's time is often used for
the relative pronoun. The whole piece is interesting.
79. BOOLE TO DE MORGAN, 6 NOV. 1862
I^have been studying a bundle of letters of yours on Logic, and now
that^ I sit down to write a few lines to you feel that I have nothing to say
about them but that I have been interested in them very much. But it has
been like the reviving of an interest that had died. There is absolutely no
person in this country except my wife with whom I ever speak on subjects
like this. I feel this as one of the many drawbacks in living in this country
and as not the least of them. However I do continually look forward to a
time when I shall study Logic again, and begin to hope that it is not far
off. I do not so much care about the mere forms of Logic as about the
philosophy of the connexion between thought and speech. What do you
think for instance of E. Renan's declaration that to the Semitic mind
expressing itself in the Semitic forms of language no Logic was possible. I
wish I knew Hebrew or still better Arabic.
One of your postulates I have always felt doubtful of. It is that a term
divides 6 the Universe that therefore its local extension is the individual its
highest the penultimate one - all but one. I don't deny that it is possible
to devise a scientific scheme of logical forms on this basis. I don't deny
that common language often seems to favour the idea of such a postulate
as its ground. But this is not uniformly the case. I think the nature of the
discourse, the state of mind of the speaker determine generally not only
the extent of the Universe of discourse but also the extension of terms.
But very often misunderstandings about these things arise and must be
settled by distinct question. In what sense do you speak of All, What do
you mean by Some? I think that limiting conditions ought in general not
to be introduced in the scientific treatment of a subject. Conceptions and
terms ought to be as general as possible and the limitations introduced
afterwards when wanted. Indeed I have been led to think in pure logic the
existence of the objects ^ designated by the terms ought not to be
assumed, even the categorical proposition partaking of the hypothetical
character or being connected with a tacit hypothetical
All men (if men exist) are mortal.
Mathematically I would say that limiting cases ought to be included not
excluded.
LETTERS 75-82 103
I will look at the syllogisms of Hamilton but I shall do it for your sake
not his. He and all his followers appear to me to have been trifling when
writing about Logic. The notion that they have mapped out the whole
kingdom of formal thought is a delusion that can only exist through ignor-
ance — a kind of ignorance which prevails in no other subject.
80. DE MORGAN TO BOOLE, 7 NOV. 1862
I have not one person to whom I can speak on logic — nor, except
pupils, on mathematics. I go from one month to another without any con-
versation on my studies with a person whom I cannot claim to teach. And
you might live in London and do the same: therefore I warn you against
the notion that you are a mental Robinson Crusoe. How many are there
who can talk or think of the first principles of anything? Even those whom
one would surfeit, by reason of the depth of their knowledge for applica-
tions make a terrible hash of any attempt to probe anything to the founda-
tion.
As to Semitics not being capable of logic, the first thing that arises in
my mind is the account given me by a relation who was diplomatically
employed in Persia. He says the Persian is a stickler for the precise sense of
words in all his dealings. If he can entrap you into a phrase which is liter-
ally explicable in his own favour, he claims to hold you to it. He would, if
it suited him - for he knows better than to let the net catch himself -
bind the Spartans to keep their laws for ever, when Lycurgus had made
them promise that they would do so till he came back — he having given
them to understand he was going away for a while. Now all this — though
very crafty — is logical craft: these shufflers must have logical power — for
shuffling is of its own nature perverted logic.
I think I shall persuade you at last that the term divides the universe in
fact — and ought to do in reason. 7 I admit a chapter on the omniternal
name — and one on the vacuous name. Would you under some, include the
limiting case none! If the x of logic may = 0, then 'Some A is 2?' is always
true. With the omniternal & the vacuous terms I think you can do nothing
in comparison of relations. If, the universe being animal, I have occasion to
affirm sentience of a species, I think I am — for the moment — recognising
something external to my universe; that is, thinking in another and larger
universe. I am satisfied that A is always in thought compared with not-A
except in this one proposition: 'let us dismiss not-A' — that is, let A be
our universe; let our propositions refer to distinction between one thing
and another in A .
Is it the rule of mathematics that the limiting case is included?
To my system it matters nothing but this — when you will introduce
the idea of your universe, you introduce it as distinguished from your non-
universe: for you cannot bring in your universe without bringing in the
idea of that which is out of it. A system which admits and systematises
contrary notions brings in non-U or u, whenever it brings in U at all.
Thus
Every X is U
Every Y is U
has reference to extents out of U, or is unmeaning. And the conclusion
1 04 THE CONTROVERSY WITH HAMILTON'S SUCCESSORS
Some things are neither Xs nor 7s - refers to those things out of U.
As to the Hamiltonian — or any system — mapping out thought — they
might as well say that Columbus mapped out America. I am sure that the
forms of thought are of a development as yet unconceived — and will be
more so as we get higher.
I have my paper no. V [D 1863c] before me in MSS (from Cambridge)
for some last corrections. Mr Baynes, you will see, has not satisfied me. I
would bet even that he will not say yes or no to my question in today's
Athenaeum, without qualification. If he do, it will be the first time I have
got a decided answer about quantity from an ordinary logician, in print or
out of print. I defy you to give me a writer who is clear on the meaning
his 'some'.
Baynes's not seeing my letters is all gammon. He tells me that he saw
the first — I may think it possible that he thought it best not to see the
others. Mansel who bridled up when he thought he had some defence for
H's mathematical blunders — tells me — by way of excuse for not seeing
that I had challenged any one to say that his editor did not propound
paralogisms - that he 'does not always see' the Athenaeum.
Our kind regards to Mrs Boole — the small ones well, I hope.
81. BOOLE TO DE MORGAN, 10 NOV. 1862
Do you happen to have preserved and can you give me the date of a
letter in which I communicated to you the method of solving simultaneous
differential equations.
82. DE MORGAN TO BOOLE, 12 NOV. 1862
I consider a letter such as this a limit to append the answer: a thing
sometimes useful in priority matters. 8
The only letter having reference to differential equations which I can
lay my hand bears date February 12, 1862 and refers to
'1st A system of n — r simultaneous differential equations between n
variables . . .'
'One of the most important applications ... is to the . . . conditions under
which
Rr + Ss + Tt + s 2 - rt = V
admits of . . . integration
'Please keep this letter'
That is, I suppose, the letter you refer to.
Notes
1 A discussion of the influence of this controversy on Boole's Mathematical
Analysis of Logic, Boole 1847a may be found in Laita 1979.
2 A spicilegium is 'a gleaning, a collection or anthology' according to the Shorter
Oxford English Dictionary.
3 See Die religiose und culturhistorische Bewegung im Judenthum, Gegenwart
1848, vol. 1,253-406, vol. 10,526-603.
NOTES 105
"•The lines of Shelley are from Julian and Maddalo, written 1818-19 after
Shelley's first visit to Venice, where he met Byron (who is identified with Count
Maddalo of the poem) and first published in Posthumous Poems, 1824. Shelley
wrote 'Most wretched men/ Are cradled . . .'.
s The name of the person criticised by De Morgan is not easy to read, but I think
Jowett is the correct reading.
6 See Letter 70.
7 See Letters 70 and 79.
8 De Morgan wrote this answer to Letter 81 upon the blank portion of that letter.
7
FROM DIFFERENTIAL EQUATIONS TO
SPIRITUALISM: 1863-1864
The final group of letters span the seventeen months from January 1863 to
May 1864; Boole died in December 1864. The letters contain the usual mixture
of serious mathematical and logical matters and light-hearted general remarks.
Both Boole and De Morgan mention their being busy. Boole in Letter 83
(3 January 1863) says: 'Your letters on Logic are not forgotten and will be
taken up again in due time. I want to get clear of mathematics . . .'. De Morgan
in Letter 86 says: 'I have escaped at last from Session work ... I have put
differential equations by until I come on the subject again, in Heaven's good
time . . .'.
Among the variety of minor points we find Boole asking if De Morgan has
any spare copies of the Life of Walsh (Boole 1851/z) 'for Mrs Boole who has not
read it'. Boole apparently had no difficulty in disposing of his 50 reprints
(see Letter 38). De Morgan indicates his dislike for the German language in
Letter 86 (5 July 1863) — T am not in love with it ... I impute to that unfor-
tunate language seven deadly sins . . .'. Commenting on some of Boole's logical
remarks and a paper on probabilities, De Morgan says: 'We are, I see, on differ-
ent rails, but we may come to a junction. I feel sure of there being many separate
routes . . .'. Another form of transport crops up in Letter 85 (7 February 1863)
when De Morgan relates a 'remark of an old gentleman in an omnibus today'.
De Morgan's wish for an 'inverse method of ehmination' in Letter 85 (7 February
1 863) seems to have remained unfulfilled.
Letters 83-86
By far the most important topic discussed in these letters is contained in
Letter 84 (7 January 1863). The major part of this letter is concerned with the
simplification of a differential form. The problem is to express "Ej^Xtdxi as
2" =1 UidVf. Boole states (but gives no proof of) a number of relations which
X t , U t , V t and their partial derivatives must satisfy. These are more intelligible in
modern notation. Writing
LETTERS 83-86 4 107
etc., and using matrix notation so that
Bu = X U -X H Ki,j<2n
and Ajj are the elements of the inverse of the matrix [5 y ] the relations become
I AijXiUkj = U k (I)
lA ij V kj = (II)
X4/tfwOi/ = o (in)
I^/^«^U = (IV)
l^UkiVu = 8 W (V)
where the summations are over i and / which range from 1 to 2n. This work
arises from Gebsch's discussion of Pfaff 's problem; this problem concerns the
solution of a total differential equation in which not all the integrability con-
ditions are satisfied. Pfaff wrote on this in 1814 (Pfaff 1815). Clebsch's dis-
cussion appeared in Crelle (Clebsch 1862).
De Morgan replies in Letter 85 (7 February 1863): 'Your method will, I see,
simplify the matter, so far as I understand it from you . . . You will of course
shape it and publish it'. Boole began to write a paper in German on this work
but death intervened before it was completed; the fragment was included in the
second edition of Boole's Differential Equations (Boole 1865, 715—7).
A mention of Isaac Todhunter in Letter 83 (3 January 1863) reminds us of
another piece of work that Boole did not live to complete — the second edition
of his Differential Equations. He had planned extensive changes; after his death
I. Todhunter took the responsibility for the second edition; he made only minor
changes in the existing text, but put the major additions that Boole had been
writing in a Supplementary Volume which appeared, as did the revised second
edition of the original work, in 1865. The chapters of the Supplementary Volume
are numbered consecutively with those of the original work, viz. 19 to 33.
De Morgan, in Letter 86 (5 July 1863), writes on matters of logical termin-
ology. This letter also refers to a possible sixth paper On the Syllogism; such a
paper was never written, although he did sketch an outline of it, which was
printed by Heath in De Morgan 1966, 346-7.
Also in this letter De Morgan mentions 'a paper I have long threatened myself
with — on infinity \ The main object of the paper, which is in two parts, is to
rescue from oblivion a concept of the actual infinite — as opposed to infinity
as a description of a kind of limiting process.
108 FROM DIFFERENTIAL EQUATIONS TO SPIRITUALISM: 1863-1864
The early pages of the first part contain an interesting recapitulation by
De Morgan of the views of various philosophers and mathematicians upon infinity.
He starts with Aristotle, and proceeds with Leibniz, Locke, D'Alembert, Hobbes,
John Mill, Kant, Berkeley. Later he connects his ideas on infinity with related
ideas on infinitesimals and introduces ideas of orders of infinity. The latter is
the primary concern of the second part of the paper where what is perhaps the
most interesting idea of the paper appears. De Morgan introduces the idea of
infinite quantities of various integer orders writing A n , B n to denote two
infinites of the nth order. He then defines the notation = m to indicate that the
symbols on either side of the sign = m denote infinities which differ by an
infinite of order less than m. This idea is used also in respect of infinitesimals.
Thus
dx =° dx + (dx) 2 ,
(one of De Morgan's examples) indicates that we are able to discard higher
order infinitesimals - without disregarding the proper meaning of =.
De Morgan's wife's book on spiritual (i.e. psychic) experiences (S.E. De
Morgan 1863) is mentioned lightly as 'a funny production' in Letter 86.
De Morgan wrote the preface under the pseudonym A.B. His wife took CD.
as pseudonym. De Morgan 'proposed that the title page should have the legend
"Jack Sprat could eat no fat/His wife could eat no lean". But this was judged
infra dig\
De Morgan's account of a remark of Samuel Johnson's in this letter lacks
the force of the original. On being asked by a Scot what he thought of Scotland,
Johnson replied: "That it is a very vile country to be sure, Sir'. 'Well, Sir', replied
the other, 'God made it'. 'Certainly he did', (answered Johnson), 'but we must
always remember that he made it for Scotchmen, and comparisons are odious,
Sir, but God made hell'; (Hill 1897, vol.1, 265).
83. BOOLE TO DE MORGAN, 3 JAN. 1863
I send you a paper in another cover. Do you happen to have one or
two copies of the Life of Walsh [1851fc] to spare. If one please send it to
me for Mrs Boole who has not read it - if two please send one to me and
one to Mr Todhunter of St John's College Cambridge if you do not think
it too much trouble.
I wish you all, and Mrs Boole joins me in this, a happy new year. Your
letters on Logic are not forgotten & will be taken up again in due time. I
want to get clear of mathematics but this will not be till a new edition of
the Differential Equations is out, and certain portions of this subject have
been quite changed in the year or two which have elapsed since the first
edition was published.
84. BOOLE TO DE MORGAN, 7 JAN. 1863
What I now write will interest you a good deal or not at all according to
whether you have ever been working in the same direction or not.
LETTERS 83-86 109
Clebsch in Crelle vol.60 Heft 3 [Clebsch 1862] has made a great step in
the theory of Pfaff 's problem of reducing
X l dx 1 + X 2 dx 2 . . . + X 2n dx 2n
to the form
U 1 dV 1 + U 2 dV 2 ...+ U n dV n
He has found the partial differential equations which V x . . . V n satisfy.
On reading his paper it occurred to me that a method founded on the
Calculus of variations, which I had applied to some other problems for
instance to the proof of Jacobi's principle of the Ultimate Multiplier ought
to give the general solution of Pfaff's problem. I tried & succeeded
deducing the partial differential equations for U t , . . . U n as well as for
V x , . . . V n ; as well as also those connecting the two sets of quantities.
They are as follows
dX h dX k
1st Let — = (h,k)
dx fe dx h
Let the linear algebraic equations 1
(1,1)^ + (2, l)s 2 ... + (2n,l)s 2n = t x
(1, 2)Si + (2, 2> 2 . . . + (2n, 2)s 2n = t 2
(1, 2n)s x + (2, 2n)s 2 . . . + (2n, 2n)s 2n = t 2n
be solved determining s\, s 2 . . . s 2n and let
s r = A r \t x + A r2 t 2 . . . + A r2n t 2n
be the type of the solutions. The whole series of quantities A^ is thus
determined as functions of jc 1} x 2 . . . x 2n .
Then 1 st any quantity Cj satisfies the partial differential equations
Zri dUs
I^feX,,— L = U t (I)
h fe dx fe
h & k each admitting all values from 1 to 2n inclusive
2nd Any quantity V t satisfies
ZZ^hfeT" 1 = (II)
h fe dx fe
3rd Between any two U if Uj exists the relation
^ Ahk dViT = (m)
h fe ax h QXk
4th Between any two V t , Vj the relation
v,^ dV t dVj , x
ZZ^hfeT^Tf=0 (IV)
h k ax h O^fe
110 FROM DIFFERENTIAL EQUATIONS TO SPIRITUALISM: 1863-1864
5 th Between any two U u Vj the relations
\k Ah *dx h dx k 0ifinot=/ ^
(II) & (IV) are what Clebsch gets. His analysis is of the most extraordinary
complexity. What I send this to you for is chiefly to illustrate by a result
the remarkable separating power of the calculus of variations in other
modes of integration giving in an orderly series the results of complex
transformations practically all but unmanageable by other modes.
A happy new year to you all from me & mine.
85. DE MORGAN TO BOOLE, 7 FEB. 1863
I thank you for yours of 7 Jan. Your method will, I see simplify the
matter, so far as I understand it from you. But it is not simple in itself.
You will of course shape and publish it.
I have not thought about differential equations for some time. I wish
somebody would study the inverse method of elimination: that is, the
an ft'-reduction of one equation to two or more with introduced letters.
For example
<P(x,y,y' ...y in) ) =
that this shall result from
F(x,y,...y w ,v) =
f{x,y,...y in \v) =
differentiate these k times giving 2k + 2 equations containing x,y, . . .
y (n+k)^ Vf v ,. . . lt k \ Let 2k+2 = n + k+2 or k - n[ ;] eliminate
y y' . . . j (2n) from the 2n + 2 equations, and there remains an equation
between xvv' . . . if- n) .
If this can be integrated, substitute for V in F = 0, /= 0, and 2n~2
of the differentiations, which will then be between x y y' . . . 7< 2 " _1 >.
Eliminate / , y", . . . y^ 2n ~ l) from 2 + 2n — 2 equations or 2n equations,
and y is found in terms of x.
[5 lines are crossed out here.]
I meant to work out an example, but I find I must stop. I have a pile
of letters to answer. With our kind regards to Mrs Boole.
[P.S.] Remark of an old gentleman in an omnibus today
'These omnibus conductors must be the happiest fellows in the world.
They never say anything but 'full inside' and 'all right'.
86. DE MORGAN TO BOOLE, 5 JULY 1863
I have escaped at last from Session work and several adjuncts, and
remember that I have to thank you for letters and print. Your German
paper [B 18636] I can just manage. As to that language I am not in love
with it enough to learn more of it. I am tired even of the introductions
LETTERS 83-86 111
from it into English. I am much inclined to copy Samuel Johnson's
sarcasm on the Scotch 'I cannot imagine, Dr. J. why God Almighty made
Scotland!' Why, Sir! you are to remember that he made it for Scotchmen."
Very well, then German was made for Germans. I impute to that unfortu-
nate language seven deadly sins, which are as follows —
1 . Too many volumes in the language
2. Too many sentences in a volume
3. Too many words in a sentence
4. Too many syllables in a word
5. Too many letters in a syllable
6. Too many strokes in a letter
7. Too much black in a stroke.
I have put the differential equations by until I come on the subject
again, in Heaven's good time. I don't know whether you find what I do,
namely, that my subjects of thought are not self-selected: they come
because they must. There is 'whatever is, is,' which settles the present, and
che sara sara which provides for the future, and c'est egal which prevents
any wish to alter the arrangement.
As to the logic I had hoped that no. V [D 1863a] would end it: but
no. VI begins to loom in the distance. Interposed, however, is a paper I
have long threatened myself with, - on infinity [D 1864a]. I hope to
succeed in differing from every body most completely.
Your last logical remarks I must omit till I have read your paper on pro-
babilities more fully [B 1862a, <2?]. We are, I see, on different rails, but we
may come to a junction. I feel sure of there being many separate routes, or
which appear separate, at present. But when a paper is just finished, I find
a certain inertia about me as to the whole subject. I suppose I am like a
gorged boa constrictor.
I will add one little simplification which shows that common words
may be made very useful.
When two universals come together or two particulars, let us say we
have a level When a universal and a particular, let us say we have a slope.
Universal followed by particular, descent
Particular Universal ascent
Thus (( )) is a balanced level
(( (( is an unbalanced level
(.( )) is a balanced ascent
(.( (( is an unbalanced ascent
The law of the secondary relations is as follows
1 . ( ) ).( may connect
Any universal
Any unbalanced slope
2. ) ( (.) may connect
Any particulars
Any unbalanced slope
3. )) (.( may connect
Any balanced level
Any descent
112 FROM DIFFERENTIAL EQUATIONS TO SPIRITUALISM: 1863-1864
4. (( ).) Any balanced level
Any ascent
1 ' M()( ' ) ( )VoV) etc - arevaud
2. ()(.))•) ))(•))( are valid
3. ()))(.( ).())() are valid
))))((
4. ()(((.( ).(((()
))((((
I am not sure whether in some of your remarks you are quite aware of
the change which there is in my use of the word mathematical, from and
after my third paper. Formerly, I should have called the numerical syllo-
gism mathematical, as opposed to the ordinary one; but latterly all the
logic of extension is mathematical - as opposed to the forms of intension,
which are metaphysical. Thus man and brute are parts of animal, mathe-
ematically: animal and reason are parts of man, metaphysically. But this
distinction is postponed in the latter part of my last paper.
I have been lately engaged in quite a different kind of job. My wife has
collected all the spiritual experiences of her own and others, and has made
thereof a book and an argument, by CD. I have written a preface, as A.B.
contending that though facts are true, the spiritual hypothesis is too hasty,
and assailing the philosophers for their omniscient mode of decrying facts
by the light of nature. The two will appear together in a few weeks, under
the title of 'From Matter to Spirit'. [S.E. De Morgan 1863] I proposed that
the title page should have the legend
Jack Sprat could eat no fat
His wife could eat no lean
But this was judged infra dig.
Letters 87-90
The last letters contain little of substance. Nevertheless the letters show how
much each valued the exchange of ideas provided by their correspondence. Thus
Boole in Letter 89 (3 May 1864) says: 'I was glad to see the well-known hand-
writing again'.
Letter 87 (8 August 1863) is an answer by De Morgan to a query raised in
letters that have not survived. On this occasion the loss makes the remark
barely intelligible, but the query appears to relate to an actuarial problem.
In letter 87 the mention of his wife and children being at the Welsh coastal
resort of Port Madoc brings to his mind a line of S.T. Coleridge:
And delights in the things of earth, water and skies;
LETTERS 87-90 113
this line comes from Metrical Feet, Lesson for a boy, which was written for
Hartley Coleridge about 1806, and first published in 1834.
In Letter 88 (25 April 1864) De Morgan announces a theorem on divergent
series — a subject which had interested him for 20 years; in 1844 he published
a thoughtful paper (1844c) on it. He wrote a paper (1864ft) giving this theorem
as well as other discussion about divergent series and subsequently two other
papers containing related ideas (1865a, 1868c).
The substance of the theorem is this: if a series S(— Vfa z , where eacha 2 is
a non-negative function of some variable, is convergent for jc<jc (or >x
perhaps), a z -* 1 as x ->• x and
then
and hence
lim -^ = 1 as z -*■ °° and x -* x
lim (X(-l)*a 2 U limf^^W
The genesis of this idea is presumably the series 2(— Vfx z , 0<x<l. De
Morgan's objective in stating this theorem is the problem of the meaning of the
divergent series 1 — 1 + 1 — 1 . . . The series 2(— lYa z has (in De Morgan's
phrase) the 'limiting form 1 - 1 + 1 - 1 . . .' (De Morgan 1864ft, 191).
The proof offered by De Morgan is hardly satisfactory. By appeal to Taylor's
theorem applied to a z ,a z+2 as functions of z he obtains
« 2+ i = a z + a' z + ka'Uv < v < 1
a*+2 = a z + 2a' z + 2a" z+tl < ju < 2
az—<*z+i _ 1 a'z + Wz+v
o z ~a z+2 2 a z +al +tl
But, De Morgan argues, a z+v , a' z+fJL are infinitesimal compared with a' z when z is
sufficiently great, so 'at the limit of summation'
az + gg+i _ I
a z -a z+2 2
(There are two or three misprints in the argument as printed in the paper, so I
have here not quoted De Morgan verbatim.) Following this argument De Morgan
comments: 'The above paragraph will, I hope, be narrowly scrutinized . . .'
(1864ft, 193). Also he gives a geometrical form of this argument — which indi-
cates the requirement that a z must be meaningful for all positive z, and that a z
tends monotonically to zero as z -*■ °°.
The weakness of this argument will be apparent. And it is significant that his
later paper, 'Note on 'A Theorem relative to Neutral Series' in Vol. XI, Part IF,
begins with the words 'The detection of an inaccuracy in my paper on neutral
1 14 FROM DIFFERENTIAL EQUATIONS TO SPIRITUALISM: 1 863 -1 864
series led me to think again on the subject' (1868c, 447). But these later
thoughts included no more convincing results than had appeared in the earlier
paper.
Letter 90 bears no indication of either date or addressee: it is in De Morgan's
hand, and it is included in the packet of letters from De Morgan to Boole in the
collection MS Add 97. One can only infer that it was possibly addressed to Boole.
It shows De Morgan in typical vein, combining a mathematical enquiry with an
experimental use of a patent writing instrument.
87. DE MORGAN TO BOOLE, 8 AUG. 1863
I think your plan is right. I thought the best way to try it was to do it
my own way first - and I find we agree, with little differences of method.
If we suppose the number kept up to 1 39 by substitution of new living for
dead as fast as they die, we certainly are in a ± state of ignorance as to
what would be the effect of assuming 167/(365 X 139) as the fraction of
daily mortality.
My wife etc. are at Port Madoc in Carnarvonshire - very happy in the
'things of earth, water, and skies'
as the rector of Opium-cum-metaphysics said. And I am here as usual,
routing in my book like a pig in a potatoe garden, who does not need
much care where his snout goes, as he is sure of finding something.
CD. and A.B. (praefator) are printed off and will appear in October.
It is a funny production. The preface writer makes cock-shies of the
philosophers - and is rational, so far as his gravity will allow towards the
people who know what is and is not a priori.
I hope Mrs Boole and the children are well.
88. DE MORGAN TO BOOLE, 25 APRIL 1864
I send you a theorem which throws some light upon a difficult point.
Let a — a i + a 2 ~~ ■ ■ •
be the limit of summation of a converging series which continues con-
vergent up to a limit of variation of the terms, in which it is lost to calcu-
lation in the form 1 — 1 + 1 — 1 + ...
First, let the law of the terms be, or finally become, permanent: that is, so
far as this, that a z+t /a z approaches at last permanently to a fixed limit (of
course not > 1 but 1 at the limit of variation, if not before)
The the limit of variation of a — a x + a 2 — is the limit of (a z — a z+ 1 )/
(flz —<*z+2)
And this is always |.
Secondly, let the law of the series proceed by cycles of even number of
terms, so that
a 2nz ~~ a 2nz+A^ a 2nz+l ~ a 2nz+2\->l
a 2nz+2n-\ ~~ a 2nz+2n
LETTERS 87-90 115
approach in ratio to k , k x . . . k 2n -\ ■
Then the limits of variation of
a — a 1 +a 2 ~... and ]
J where sum is 1 at last
a 1 -a 2 + a 3 — . .. J
have the ratio of
k + k 2 + k 4 + . . . + k 2n - 2
to
k x + k 3 + k 5 + . . . + k 2n -i
But if the number of terms in a cycle be odd, both series have \ for the
limit.
This theorem explains every case I ever met with in which 1 — 1 + 1 —
1 + . . . was alleged to mean other than 2-
I hope you are all well. Our kind regards to Mrs Boole.
89. BOOLE TO DE MORGAN, 3 MAY 1864
Thank you for your letter. I was glad to see the well-known hand-
writing again. I have nothing to say about the results — nothing more than
that I am glad to see you still working and that I have no doubt they are of
value. But I have no critical observations to make.
I was lecturing the other day on Spherical Trigonometry and was struck
with the cumbrous character of the proofs of Napier's Analogies. Thinking
the matter over yesterday evening I was led to the following proof.
2 . / sinfa — b)sin(s — c)
1st Since tan 5 >1 = V '.
we have if m
sin s sin (s —a)
I sin (s — a) sin (s — b) sin (s — c)
tan 2 B = — etc.
sin (s — a) sin (s — b)
Hence by substitution
tan ^ + tan \B _ sin (s — b) + sin (s — a)
tan 2 A — tan \B sin (s — b) — sin (s — a)
or
sin^U + B) _ tan 2-c
sin^U— B) tan^Ca — b)'
2nd By the same substitution
(I)
1 —tan \A tan \B _ sin (s — a) sin — b)
1 + tan 2"^ tan 5 £ ^2
1 +
sin (j — a) sin (s — b)
116 FROM DIFFERENTIAL EQUATIONS TO SPIRITUALISM: 1863-1864
1
2
(4
l —
sin (s — c)
sin s
1 +
sin s
sin (s — c)
sin s
— sin (s —
c)
cos
sin s
+ B)
+ sin (s —
tan \c
c)
(II)
" cos sU- 5) tan £(« + &)'
Is this new?
My kind regards to AB or CD I forget which Mrs De Morgan is. I read
the preface and the book - through, which will show you that you had not
failed to produce interest. But I was not convinced. I do not at all under-
stand why the reality of the phaenomenon of a table rising up from the
ground and remaining suspended in the air without any hand touching it
or material communication of any kind (which I have heard positively
asserted by a person who said he saw it with his own eyes) should not be
investigated by scientific men as any admitted or presumed natural
phaenomenon would be. I confess my opinion to be that the exhibitors
of such phaenomena dare not submit them to such a test. One of three
suppositions must be taken to represent the real state of the case. Either
such phaenomena are done by juggling i.e. by unseen mechanical appliances
dependent upon known laws, or by the application of unknown laws of
nature, or by agencies which are not in the ordinary sense of the term
natural. As to the second supposition which would involve that Mr Home
knows more of natural laws than all the world of scientific men, it may
I suppose be put aside. I know of no admitted phaenomenon having any
kind of analogy to the suspended table but that of the suspended needle
in a helix through which a galvanic circuit passes - but that is an experi-
ment of the most refined and difficult kind. We are I think then shut up
to juggling or to influences not natural in the ordinary sense of the term.
I say in the ordinary sense of the term because there may be properties
of the bodies & souls of living men that are so different from anything in
admitted physiology or psychology that they must appear at first to be
out of the range of natural things. I don't say there are. I suspend my
judgment. But to return to what I first said why do not the spiritualists
set themselves clean before the world by inviting scientific men to examine
the physical phaenomena in their own way?
P.S. My wife expressly sends her compliments to AB and CD.
90. DE MORGAN TO [BOOLE?], UNDATED
Herewith a copy of the paper which you need not return. Have you any
reference to any good writing on symmetrical functions of the root of
unity? If m/ ,
a,b,c,. . . be VI *
a u b u c u .. . be "tyl's
etc.
LETTERS 87-90 117
I want an easy way of finding a symmetrical function of the form
I have a faint remembrance of having once had a rule to distinguish which
are zeros and which are integers. But if so, I have forgotten it
What I am writing with purports to be not pencil but solidified ink. It
is said not to rub out, after the first hour. It is called 'Melvilles Patent'.
[Two large crosses appear here in the letter.]
The one on the right has had 25 hard rubs with a bit of Indian rubber as
soon as written.
Notes
The notation (2, l)s 2 stands for a coefficient multiplied by s 2 : today we should
write a 2 is 2 , perhaps.
In the following formula Boole wrote 'sin a' where 'sin (s — a)' should appear.
I have corrected this minor slip.
8
EPILOGUE
Boole died in December 1864 at the early age of 49. He left a widow and
five daughters and De Morgan was active in canvassing support for an appli-
cation to the Government for a pension for them. A draft letter to this end
survives (De Morgan 1864, other manuscripts). In addition to throwing light
on the parlous financial position that Boole's early death left his wife and
family in, this draft letter contains an assessment by De Morgan of Boole's
work.
De Morgan begins by saying that Boole worked for 15 years as a public
teacher, the first thirteen of these with small remuneration, so no provision
for his family was possible. Boole was, De Morgan said, a success as a teacher,
was held in high regard by students and colleagues for his character and intellect
generally (De Morgan's emphasis). He continued:
We submit that one who has done so much, and has worked through the period
of comparative failure and discouragement, may without presumption, be
presented to H.M. Government with good hope of favourable consideration.
In addition to Boole's record of public service De Morgan placed on record his
intellectual achievements:
He is one of those men who have devoted rare genius with great success to parts
of science which have no reward except what comes directly from the public
purse, or else to men of academical education from the endowments of their
Universities.
De Morgan gave some brief biographical notes: Boole was self-educated, a
schoolmaster; then he became known through the papers contributed to the
Cambridge Mathematical Journal - papers which contained
some very remarkable speculations which can here be described only in general
terms, as extensions of the power of algebraic language. These papers helped
to give that remarkable impulse which algebraic language has received in the
interval from that time to the present. Various papers followed, one of which
received the gold medal of the Royal Society.
EPILOGUE 119
After mentioning the value of the texts that Boole had written De Morgan
referred to his work in logic:
That peculiar turn for increasing the power of mathematical language which is
the most characteristic point of Dr Boole's genius, was shown in a singularly
remarkable way in his writings on logic. Of late years the two great branches of
exact science, mathematics and logic, which had long been completely separated,
have found a few common cultivators. Of these Dr Boole has produced far the
most striking results. In alluding to these we do not say that the time is come
in which they can ever be generally appreciated, far less extensively used. But if
the public acknowledgement of progress and of genius be delayed until the
whole world feels the results, the last century, which had the benefit of the
lunar method of finding longitude, ought to have sought for the descendents
of Apollonius, to reward them for his work on the conic sections.
There followed a deleted portion which attempted to explain in simple terms
the significance of Boole's
system of logic [which] shows that the symbols of algebra ... are competent to
express all the transformations and deductions which take place in inference.
De Morgan also recorded Boole's honorary degrees (from Oxford and Dublin)
and mentioned that there was:
a prospect of admission into the French Institute, cut short by his death.
He concluded the letter with the hope that the grounds he has put forward
'are good and our request worthy of favourable consideration'.
After Boole's death his wife presented his manuscripts which related to
mathematics and logic to the Royal Society. In 1867 De Morgan examined
these to ascertain whether there were any that merited posthumous publication.
Apart from Boole 1868 De Morgan found nothing complete enough to warrant
publication. He reported:
After much consideration I am satisfied with two things. First, the author
himself, would have objected to their publication as they stand. He would
have introduced much change of expression and allusion to his higher views,
or rather, preparations without allusion.
Secondly, a false impression would be produced: a posthumous work by George
Boole on logic would be taken for his latest and highest view. Those who
would know better when they came to open the book would not find out
how the matter stood, would really believe they were in possession of all Boole's
intentions. And as a hundred copies would sell for one of the laws of thought, a
very wide misapprehension of the contents of the 'Laws of Thought' would get
about. [De Morgan 1867, other manuscripts].
120 EPILOGUE
De Morgan was 58 years old when Boole died; he had a little more than six
years left to him, but the fires were burning low.
In October 1864 De Morgan's second son, George Campbell De Morgan,
together with a friend, A.C. Ranyard, conceived the idea of a mathematical
society at University College. This idea developed into the London Mathematical
Society which first met in January 1865. De Morgan was the first President and
delivered an inaugural address (De Morgan 1 866g). 1
From this time professional and personal matters were a source of un-
happiness for De Morgan. In September 1865 two friends of long standing died:
W.R. Hamilton (1805-65) and W.H. Smyth (1788-1865). Also in 1865 the
dispute at University College concerning the College's failure to appoint the
Rev. James Martineau to the Chair of Mental Philosophy and Logic erupted - he
was clearly the best candidate but was a notable Unitarian. De Morgan concluded
that the real reason for Martineau's rejection was his religious beliefs and this
he considered to be contrary to the strictly non-sectarian constitution of the
College; he resigned in protest. According to his wife:
My husband told me that during the session in which he worked after his
resignation was sent in he met his colleagues as before in the Professors' room.
Not one of them ever spoke on the subject of his retirement, and he left
the place without one word of acknowledgement for all he had done for it
[S.E. De Morgan 1882,358.]
De Morgan's earlier resignation - in 1831 - had been as a protest over the
action of the Council in the dismissal of G.S. Pattison, the Professor of Anatomy.
Two other professors had resigned with De Morgan; F.A. Rosen, the Professor of
Oriental Languages and G. Long, the Professor of Greek.
In October 1867 George Campbell De Morgan died, followed in August 1870
by his sister Helena Christiana, both victims of tuberculosis. De Morgan himself
was ill in 1868.
De Morgan's papers in his last years, i.e. 1865-71, were often short notes
of little importance. In these years he contributed a number of notes to the
Assurance Journal and the Journal of the Institute of Actuaries. His longer
papers continued to appear in the Transactions of the Cambridge Philosophical
Society - in particular De Morgan 1866c, 1868c - but these papers were only
a reworking of earlier ideas. He also contributed two brief notes to the early
Proceedings of the London Mathematical Society (De Morgan 1868a, b), but
of some historical interest is the report in volume 1 of the Proceedings of his
speech at the first meeting of the Society (1866g).
The letters have shown De Morgan as a writer with a humorous manner. One
feels that he must have been a valued acquaintance. A person who recorded his
appreciation of De Morgan's friendship was Henry Crabb Robinson, who said
of De Morgan:
EPILOGUE 121
He is the only man whose calls, even when interruptions, are always acceptable.
He has such luminous qualities even in his small talk. [Robinson 1872, vol. 2,
385.]
An autobiographical note and his will reveal De Morgan's view of himself:
first, professionally:
Mr De Morgan is one of the few mathematicians who hold mathematics to be no
sufficient substitute for the study of logic ... He has been a voluminous writer
on many branches of mathematics [De Morgan 1860, other manuscripts. ]
Second, he gave in his will the reason why he had consistently refused to speak
about his religious beliefs:
I commend my future with hope and confidence to Almighty God; to God the
Father of our Lord Jesus Christ, whom I believe in my heart to be the Son
of God, but whom I have not confessed with my lips, because in my time such
confession has always been the way up in the world. [S.E. De Morgan, 1882,
368.]
Any comparison of the manner in which Boole and De Morgan tackled the
mathematical and logical problems they chose to attack appears to show that
Boole had the finer mind. He chose interesting and important topics, had new
ideas to express about them, and communicated these incisively. De Morgan,
on the other hand, often wrote at great length without quite reaching the heart
of the matter, although what he had to say contained interesting — often vividly
expressed - remarks. All of De Morgan's texts had short lives with the exception
of the Elements of Arithmetic (1830a), which was regularly reprinted for more
than 40 years. Only his Budget of Paradoxes (1872), a compilation posthumously
issued by his wife, has been reprinted in recent times. At least three of Boole's
books have survived. Laws of Thought (1854a) and The Mathematical Analysis
of Logic (1847a) have been reprinted in recent years. So have his texts on
Differential Equations (1859) and Finite Differences (1860); the former seems
to be in steady demand at the author's university's library.
What does the correspondence add to our knowledge of Boole and De Morgan
as persons and as scholars? The wide range of literary interests, and their language
skills, are a salutary reminder of changes that have taken place in education and
educational achievements over the past century. Of course Boole and De Morgan
were not average persons; by different routes they became exceptionally well
educated. But making comparisons between well-educated persons of today and
Boole and De Morgan, one cannot but observe that the latter had a better general
education than comparable persons of the 1970s. One would have to search very
hard among present-day university professors to find one like Boole who,
beyond the level of an elementary school instruction, had largely taught himself.
122 EPILOGUE
At this point we might usefully sum up and compare the contributions of
Boole and De Morgan in the field where, in all probability, their major contri-
butions to knowledge lie — i.e. logic.
De Morgan's early work (i.e. before 1859) on logic can be characterised as
(usually) notable extensions of the theory of the syllogism. De Morgan intro-
duces new forms of the syllogism but his ideas, though new, can still be seen as
belonging to the style of logical reasoning that began in ancient Greece. The
symbols he introduced to express the old and new types of syllogism were in
their conception not algebraic — he used literary symbols such as brackets ( , )
to denote the relationship between the statements that formed the syllogisms.
He found that his symbols could be manipulated in a way that resembled the
familiar method of manipulation of algebraic formulae (see, in particular,
Letter 70). His later work is more original and is perhaps his most lasting
contribution to logic. In the paper On the Syllogism IV (De Morgan 18596)
he moves beyond the syllogism to investigate the theory of relations. 2 Although
he was not the first to study relations in logic, he was probably the first to give
this subject concentrated attention (Kneale 1962, 427). Later Peirce was to take
up and develop the theory of relations (for the beginning of his work on the
subject see Peirce 1 870).
Boole's work on logic is thoroughly algebraic in character from the beginning.
His boldness in the adaptation of ordinary algebra to a form suitable for the
expression of logic is remarkable. Almost at the beginning novel algebraic laws
arise, e.g. x 2 = x. In the formulae by which he expands elective functions (e.g.
<j>(x,y) = xy<K0, 0) 4- x(l -yW , 0) + (1 -*)y<K0, 1) + (1 -x)(l - J>>K0, 0))
he finds it necessary to use the symbols 0/1, 1/1, 1/0, 0/0. The latter two of
these are inadmissible in ordinary algebra. But these novelties are accepted and
woven into the system by Boole. The use of algebraic formulae to express logical
problems allows Boole to enunciate and solve some that are of such complexity
that their treatment in the traditional language is decidedly difficult - see
for instance his analysis of the definition of annelida (Boole 1854a, 144—6).
Although it is not possible to find Boolean algebra, the theory of lattices or the
present-day mathematical development of propositional logic in Boole's work,
the germs of ideas which were to lead to these topics are clearly present. It was,
however, left to later workers to recognise and develop these ideas.
In a recent paper van Evra has made a reassessment of Boole's contribution
to the theory of logic. He describes Boole's work as 'one of the first important
attempts to bring mathematical methods to bear on logic while retaining the
basic independence of logic from mathematics' (van Evra 1977, 374). While
acknowledging that Boole's logic is imperfect, van Evra notes that 'he displays
a greater awareness of logic as an independent discipline, as well as more
sophistication in the use of mathematics as an aid to logic while avoiding
the conflation of the two areas, than did his successors in algebraic logic',
[van Evra 1977, 365.]
EPILOGUE 123
The long-lasting exchange of ideas in the letters leads one to consider the
question of the influence each might have had upon the thoughts of the other.
In the earlier years of the correspondence the relatively greater maturity and
experience of De Morgan compared with Boole naturally suggests that the
influence flowed from De Morgan to Boole. Examination of the letters, however,
suggest that Boole's ideas were almost entirely his own. Nearly always we
observe De Morgan reacting to Boole's ideas, rather than vice versa. Constantly
Boole is raising ideas and putting them to De Morgan for comment; further
De Morgan's reactions are not of a kind that suggest that De Morgan is capable
of directing Boole's ideas into a significantly different course. It appears^then,
that any influence that De Morgan had on Boole was either slight or delayed,
so that no immediate indication of it shows in the letters. I would go further
and claim that after the early letters the evidence of the letters shows that it was
Boole who had the most original ideas and most vigorous intellect. Consequently
if there is any question of one influencing the other in the period of Boole's
intellectual maturity it is Boole who is influencing De Morgan. However, what
might have been important was not so much the way that each directly influenced
the other as the stimulation of a steady interaction of ideas — and this stimulation
perhaps acted on a level below that of immediate overt influence. 3
It is fitting to conclude with words used by De Morgan which expressed in a
vivid manner his view of Boole's work:
When the ideas thrown out by Mr Boole shall have borne their full fruit, algebra,
though only founded on ideas of number in the first instance, will appear like a
sectional model of the whole form of thought. [ 1 S6lb, 346.]
Notes
1 For further information of the De Morgan's activities in the foundation of the
London Mathematical Society see Collingwood 1966.
2 For a recent discussion of De Morgan's theory of relations see Hawkins 1979.
3 The various influences — including De Morgan's — on the genesis of Boole's
ideas of logic are discussed in Laita 1977, 1979, and 1980.
APPENDIX:
BOOLE'S THEOREM ON DEFINITE INTEGRATION
The theorem that Boole discussed in Letter 10 (8 Jan 1847) is given in more
generality in his two published notes (Boole 1848c?, 1849a). In these he claims
that
r f(x~R)dx = [ f(v)dv (1)
where R is a rational function of x such that the roots of the equation
x — R = v are real for all values of v for which f(x) does not vanish. Although
the result appears, at first sight, unlikely the 1849 paper contains a number of
verifiable examples which suggest (at least) a substratum of truth. In fact subject
to an additional condition upon R and assuming, as Boole did, that there are
no problems with the convergence of the integrals, the result is substantially
correct. The additional condition is this: that R(x) = p(x)/q(x) where the
degree of p(x) is less than the degree of q(x) and that x —R(x) is piecewise
increasing i.e. increasing in each of the intervals into which the x-axis is divided
by the zeros of q(x). That Boole was aware to some extent of this added con-
dition is clear in that in both the papers after stating the result in terms of a
general rational function R he gives also a 'particular form'
r/f-f^) d * =/-■>> <■*•
where X { , 1 < i < n, are any real constants and a u 1 < i < n, are positive. It is
easy to see that this particular form does satisfy the condition imposed above.
The simplest version of the theorem is capable of an elementary proof: this
a
version asserts that if a > and v — x , then
x
\_J(v)dv ={_ oo /Ix-Mdx.
Following Boole I shall assume that the function is such that the infinite integrals
are convergent. First assume, further, that /is an even function. The substitution
a
v = x yields
APPENDIX: BOOLE'S THEOREM ON DEFINITE INTEGRATION
125
Fig. 2
Put
/>>*- - n 4~*)( i+ ?h -j."4~*
IS
'£*•>*-£ 4 "x)( I+ ?) dx -
7=a/jcinJ /Ijc Jdx;
-\[l + - 2 )dx.
(2)
one obtains
otH^^-'H)^
(3)
as / is an even function. The result now follows from (2) and (3). But the result
is certainly true if / is an odd function (it becomes = Oh so writing f(x) =
?(f(x) +/(—*)) + k(f(x) —f(—x)), i.e. expressing / as the sum of an even and
an odd function, the result now follows.
I turn now to the general result, viz.
[> d -C/fi^) d ^
(4)
where a;>0 (1 <z<«). The elementary approach used previously does not
seem to be susceptible of generalization. However a proof can be given on the
following lines; to simplify the details I shall take the case n — 2, but the method
is quite general.
1 26 APPENDIX: BOOLE'S THEOREM ON DEFINITE INTEGRATION
Note first that v = x — at/ix —\i)—a 2 /(x — X 2 ) has a graph of the form
shown in Fig. 2. Now consider the special case
fix) = X[x,x + 6] (8>0)
where as usual x denotes a characteristic function. Then /_" f(y)dv = 5. Also,
because of the chosen form of/
r4-^v--T) d * = £<<
J-oo I x—\i X—A2J i
where /,• (1 <z<3) denote the lengths of the three intervals in which x —
ai/(x — Xx) — a 2 /(x — X 2 ) takes values between X and X + 5. If X = x —
ai/(x — X l ) — a 2 /(x—X 2 ), then on clearing fractions and simplifying one
obtains a cubic equation
x 3 -x 2 (X l + X 2 + X)~X(-a l -a 2 +X l X 2 + X l X+X 2 X)
+ a\\ 2 + a 2 Xi — XiX 2 X = 0.
The left-hand end of the three intervals are the three roots of this cubic
equation. Similarly the right-hand ends are the three roots of the corresponding
cubic equation in which X + 5 replaces X. Thus l x + h + h is found from the
difference of the x 2 coefficients i.e. l\ + l 2 + h = §• So
("4*-- *v— vW- 8 - f"/wd..
J-oo"' ^ x-X! x-X 2 J J-°°
and the result is proved. It is now clear in what circumstances the general case
may be proved: we require that / be a function that can be suitably approxi-
mated by a linear combination of characteristic functions S x J^X[X|,x^6/]-
'Suitably' implying that a sequence of such approximations can be found so that
their integrals 2^ Y t S t shall have /_" f(v)dv as limit.
a
As noted above the elementary approach which succeeds for v(x) = x
does not appear to generalise. One obtains without difficulty x
J-oo J-oo I x~Xi x—X 2 J\ (;
01 +^r-,\dx
[x-XO 2 (x-X 2 )
but I have found no substitution that will reduce
r_'f-^-^)((7^ + (7^?) dx <s)
to 2/°^ f(v)dv. However the above proof indicates that (5) must indeed be
2 /_°° ~f(v) dv — and indeed in general that
In the most general form of the theorem Boole states that
{_/(») dt» = $_„ f(x-R)dx,
dx.
APPENDIX: BOOLE'S THEOREM ON DEFINITE INTEGRATION 127
where ( R [is] any rational fraction, a function of x ... and if the values of x
given by the equation x — R = v are real for all values of v included within the
actual limits of integration'. (Boole 1849 14-15).
It is rather difficult to see what Boole means by this condition. As mentioned
above the actual requirement is that v = x —R(x) should be piecewise increas-
ing. Thus R may be a sum of terms of the form c,/ (x — X,) ', where t,- is an odd
(positive) integer and a { > 0.
I turn now to Boole's proofs of this result. In the announcement in
Liouville's Journal there is no indication of proof. The proof in Boole 1849 is
slight to the point of unintelligibility:
Suppose that f(v) is discontinuous, let it be imagined to vanish for all values of v which
do not lie within the limits p and q. These are then the actual limits of integration. Accord-
ing to this definition of the character of the function /, it is evident that f(x —R) will
vanish whenever x — R transcends the limits p and q.
Letp,p 2 . . .p n be the roots of the equation
x—R = p,
and q 1 q 2 . . .q n those of the equation
x-R = q (6)
and suppose p t p 2 . . .p n and q t q 2 . . . q n arranged in the same order of magnitude, we have
then
[ 9l dxf(x -R) + f %x/(x -R) ...+ ( qn dxf(x -R) = (" dvf(v) ... (7)
JPi JP 2 JPn JP
and this may be applied to the determination of the sums of an infinite variety of trans-
cendental integrals.
The proof — perhaps more accurately described as a discussion indicative of a
proof - in Letter 10 is likely to seem obscure to a present-day reader. Note the
'deduction' of 2 dx = dv from 2 x = v. With hindsight, and a good deal of
charity, one can perceive some resemblance between Boole's argument and the
measure-theoretic one given above e.g. may not Boole's p and q be analogous to
myI + 6 and XI Although the style of argument he used seems strange now it
was common enough in the nineteenth century and one presumes was an
expression of a reasoned mathematical insight which was then generally
intelligible. We, today, may find this style strange and imprecise; but that is no
good reason for dismissing it out of hand.
Boole gives a number of examples in his paper (Boole 1 849). The first - and
one supposed the formula that led him to the general result - is that which
arises when one takes /(u) = e~" . Then Boole's theorem yields
/>*«.- r-.-H
dx
i:
* 2 dx (9)
(The numbers identifying the formulae are those given in Boole 1 849.) He also
gives the analogous result that arises from taking f(v) = e~ v , 'for n-even'. At
this point, at least, he has realised the need for / to be an even function if the
result is to be non-trivial.
128 APPENDIX: BOOLE'S THEOREM ON DEFINITE INTEGRATION
Boole claims that 'in an exactly similar manner we may deduce
J o dxcos(x 2 + ^j | o dxsin^ 2 + ^j,
but gives no further explanation. Allowing the use of complex-valued functions,
j 2
these integrals can be obtained on taking f(v) = e ; their values are, respect-
ively ;
, / n , / it
h \ ~ (cos 2a — sin 2a), h I ~ (cos 2a + sin 2d).
zosav
Another example arises on taking f(v) = — ; — -r- Boole deduces that
1 + v
r°° I r
cos \a \x
X ■ A
2 ax = 7re
1+ \x-~
X
It is a tedious exercise in contour integration to verify this odd-looking formula.
In the middle of these results (1849, 17) Boole claims 'from the known
f°° d0 0~*
integral „ — r , I in like manner deduced
jo (1 + 0)
dxjc n " 3/2 r(n-h)\/n 1
I -
Jo (a + bx + cx 2 ) n r(n)a> (b + 2\/ac) n ~>
r- dn""' _ Tin-^Wn l
Jo (a + bx + cx 2 ) n r(n)c> (b + 2y/ac) n ^
(12)'
He does not state the conditions on a, b, and c, but from the right side of (his)
equation (12) one infers that a > 0, c> and b + 2\Jac> 0. The integral he
starts from, zdd, is a form of the beta integral. The formula (12)
J o (i + 0) n '
cannot be deduced from this integral without some preliminary transformation;
it seems better to prove them by adaption of the ideas of the proof of the
general result . On making the substitution
v<
ac
N a \ c x
b + 2\Jac
in the beta integral, after some tedious manipulation one obtains
r °° -d0 = (b + 2y/ac) n - 1 *(\/al + y/cj)
a + ey
APPENDIX: BOOLE'S THEOREM ON DEFINITE INTEGRATION 129
where
J°° x r x
Va (a + bx + cx 2 ) n d *' J = V J (a + bx + cx 2 ) n d *'
a
On making the substitution y = — x in these, one finds
c
7 = /£. r ^ g * n ~" Ja = /1 r>/ s *"- 3/2 Ja
Va Jo ( a + bx + cx 2 ) n X ' *J c Jo ( a + bx + ex 2 )
Hence
.n-3/2
lo"^" 9 = <» + ^>~' (^ + ^ ^V
dx
+ bx + cx 2 y
which yields the first of Boole's formulae in ( 1 2). The other is obtained similarly.
In his Cambridge paper Boole remarked of his theorem: 'There is a fair pro-
mise of interesting if not of important consequences, but I have no intention
now of persuing the enquiry' (1849, 14). However some years later he did take
up these ideas again. The Boole papers in the Library of the Royal Society con-
tains a note which discusses this result. Although it is only an incomplete draft it
is interesting to observe in it the more mature manner in which Boole deals with
the theorem. The relevant portion reads:
One of the most general theorems to which I have been conducted may be thus stated.
It is always possible to reduce the definite integral
1>(
x-X t x-\ n J
to the form f^ o R'f(x)dx where J?' is a rational function of x similar in form to R but with
different constants.
A particular case of this theorem was given by me several years ago in the Cambridge
Mathematical] J[ournal]. It was the following:
fix) dx.
DK-v-.-^r-j:
Another particular case was described by Cauchy and has been made the subject of a
memoir. His result amounts to this: that
/*•"/(*' + £] I
* aB /l* a + ^l[dx]
may be reduced to depend on the integrals
f x 2n f(x 2 + la) dx, [ x 2n - 2 f(x 2 + 2a) dx, etc.
I have deduced Cau dry's theorem as a special result from my own.
The most general theorem is, however, the following:
130 APPENDIX: BOOLE'S THEOREM ON DEFINITE INTEGRATION
dxR(x)f(x-R) = | dxR'(x)f(x)
provided that R is such a function jc that the roots of the equation
x—R = v
are rational for all real values of v.
There are some trigonometrical functions which satisfy this condition and it is very
remarkable that such conditions have been also discussed by Cauchy. He has, if I may be
allowed the expression, discussed the condition of a problem which he had not solved.
A great many known definite integrals and sums which occur in the theory of electricity
fall under the above general results. They add largely also to our power over multiple
integrals but the new forms to which they lead are not apparently important ones.
The symbol with the interpretation which I have given to it agrees with Cauchy's
symbol of residues with this difference that it involves one additional element of mean-
ing. I have no manner of doubt that that element is of essential importance. In many cases
in which it is written as in the writings of Cauchy it requires to be supplied in a less con-
venient manner. (Boole, Roy. Soc.)
The reference to Cauchy suggests that it was in one of his memoirs that
Boole may have got the initial idea behind his result. It is not possible to be sure
which memoir of Cauchy Boole was referring to. One possibility is that it was to
Cauchy's important paper of 1823 on partial differential equations (Cauchy
1823) where some similar formulae appear in Section 11. Another paper of
Cauchy's which relates the integrals
Mn = j™ x 2n f(x 2 )dx
dx
appears in Exercises de Mathematiques Annee 1826 (Cauchy 1826). In this
paper Cauchy makes use of complex numbers to deduce such formulae as
J7* 2B e~ V 7 cos fix
Idjc.
x i
The final paragraph seems to refer back to an earlier part of Boole's note —
which is possibly lost or misplaced. The specific formulae that Boole deduced
from his theorem as examples (Boole 1849, 116—17) have an appearance which
suggests to a modern reader that contour integration might provide a suitable
means of proving them. It is possible, but not particularly convenient to prove
them by such means.
Boole's theorem is the kind of result which is discovered and forgotton more
than once. I have not seen any version as general as Boole's, or of any earlier
date. Simpler versions appear in Todhunter's Integral Calculus, (Todhunter
1857, 233) and J.W.L. Glaisher also published a note containing a simple form
of it (Glaisher 1875, 186-7).
BIOGRAPHICAL NOTES
These notes contain concise biographical information on all persons (including
fictional ones) mentioned in the letters. In the case of M. E. Boole and S. E. De
Morgan, only references other than those in terminal greetings are given.
Letter
A.B. Pseudonym of A. De Morgan. 87, 89
Ampere, A.M. (1775-1836). Ampere is best known for his
investigations on electricity, began his career with
papers on partial differential equations. 75
Aristotle (384-322 BC). Greek philosopher. 70
Barrow, Isaac (1630—77). Mathematician and divine;
sometime Lucasian professor and Master of Trinity
College, Cambridge. 64
Baynes, T.S. (1823-87). Baynes studied under Sir W. Hamilton
and later was professor of Logic, Metaphysics and
English Literature at St Andrews University. 25, 77
Beddoes, Thomas ( 1 760-1 808). Physician and writer. 64
Bertrand, J.L.F. (1822-2900). A French author who wrote
many books on mathematics and physics. 63
Billingsley, Sir Henry (d. 1606). Merchant, sometime
Lord Mayor of London. 60
Boase, H.S. (1799-1883). Boase wrote on geology and
chemistry. He was elected FRS in 1837. 66
Boethfus (c. 475—526). Roman nobleman and philosopher. 70
Boole, Mary Everest (1832-1916). Wife of George Boole
and writer on psychological and educational topics. 56, 57, 59, 61
65,69,79,83
Burgersdicius (i.e. F.P. Burgersdrjck) (1590-1635).
Dutch philosopher. 70
Caesar, G. Julius (c. 100—44 BC). Roman general and dictator. 70
Cagnoli, A. ( 1 743—1 8 1 6). Italian astronomer and scientific writer. 5 8
Carlyle, Thomas (1795-1881). Historian and essayist. 67
Cauchy, A.L. (1789-1867). Mathematician, one of the
creators of modern analysis. 60
Cayley, Arthur (1821—95). A prolific mathematician who
contributed to the theory of invariants, matrices,
n-dimensional geometry, and non-Euclidean Geometry. 53, 76
CD. Pseudonym of S.E. De Morgan. 87, 89
Chretien, C.P. (1820-75). Dean and Tutor of Oriel College,
Oxford, 1843-64. 25
132 BIOGRAPHICAL NOTES
Clebsch, R.F.A. (1833-72). Clebsch wrote mainly on the
theory of projective invariants and algebraic geometry,
but also on partial differential equations. 84
Columbus, Christopher (c. 1446-1506). Discoverer of
America. 80
Crackanthorpe, Richard (c. 1567-1624). An Oxford Divine
who wrote on logic and church affairs. 70
Creswell, Sir Creswell (1794-1863). A member of parliament,
barrister, and judge of the Divorce Court. 70
Crusoe, Robinson. Fictional character created by Defoe. 80
Davies, J.S. (1795-1851). Writer on mathematics and
science, mathematical master at the Royal Military
Academy 1834. 37, 58
Dee, John (1 527-1608). Mathematician and astrologer. 60
Delambre, J.B.J. (1 749-1 827). Historian of astronomy. 58
De Morgan, Sophia, E. (1809-92). Wife of A. De Morgan. 46, 47, 56, 57
59,61,86,87
89
De Vericour, R. (d. 1878). Professor at Queen's College,
Cork. 24, 25, 28, 47
Dickens, Charles (1 812-70). Novelist. 38
Donkin, W.F. (1814-69). Savilian professor of Astronomy
at Oxford. 33, 35, 36
Duns Scotus (c. 1275-1308). Fellow of Merton College,
and Professor of Theology at Oxford. 72
Ellis, Sir Henry (1778-1869). Principal librarian of the
British Museum from 1 827 to 1 856 4
Ellis, R.L. (1817-1859). Fellow of Trinity College,
Cambridge, a frequent contributor to, and editor
of the Cambridge Mathematical Journal. 2
Esculapius. A hero and god of Greek mythology associated
with medicine 65
Euclid (fl. 300 BC). Geometer. 60, 69
Fraser, A.C. (1819-1914). Hamilton's successor at
Edinburgh University. 77
Garibaldi, Giuseppi (1807-82). Between 1860 and 1862
Garibaldi led the revolt in Sicily and southern Italy
which resulted in their accessions to the unified
Italian state. 72
Gill, M.H. (fl. 1850). University Printer in Dublin. 46, 47
Goodacre. Not identified. 4
Graves, Charles (1812-99). Fellow and Professor of
Mathematics at Trinity College Dublin. 1 2, 24
Gregory, D.F. (1813-44). Fellow of Trinity College,
Cambridge. He was the originator and first editor
of the Cambridge Mathematical Journal. Gregory
was one of the Gregory family which included
James Gregory, 1638-75, and David Gregory
1627-1720. 3,30
Gregory, Olinthus Gilbert (1774-1841). Mathematical
master at Woolwich. 64
BIOGRAPHICAL NOTES 133
Hall, Marshall (1790-1857). A physician who was elected
to the Royal Society in 1 832. 37
Hamilton, Sir William (1788-1856). Philosopher and
logician. 11,25,64,70
74,79
Heaviside, J.W.L. (1808-97). Senior professor of
Mathematics at the East India College, Haileybury,
1838—57, and examiner in mathematics at the
University of London . 6 1
Herschel, Sir John (1792-1871). Astronomer and the son
of the astronomer William Herschel. 49, 63
Hildebert (c. 1 056- 1133). Sometime Archbishop of Tours. 47
Home, D.D. (1833-86). Home held private spiritualist
seances in London, Europe, and America. 89
Ingelby, CM. (1 823-86). Critic and essayist. 70
Isenach, J. (i.e. Justis Joducus of Eisenach) (//. 1500).
Teacher of theology and philosophy at Erfurt. 70
Jacobi, C.G.J. (1804—51). Jacobi was one of the discoverers
of multiply-periodic functions. 67, 84
Jamieson, R.A. (fl. 1860). Pupil of George Boole. 69
Johnson, Samuel (1 709-84). Lexicographer and essayist. 86
Joshua of Nazareth. Jesus Christ. 70
Jowett, Benjamin (1817-93). Professor of Greek and
sometime master of Balliol College, Oxford. 67
Kane, Sir Robert (1809-90). President of Queen's College,
Cork, from 1 845 to 1 873, and an editor of the
Philosophical Magazine. He was elected a Fellow of
the Royal Society in 1 849. 21 , 26, 32
Kant, I. ( 1 724- 1 804). Philosopher. 46
Keckermann, B. (1573-1609). Divine and author. 70
Kelland, P. (1808-79). Professor of Mathematics at
Edinburgh, 1838-79. 64
Laplace, P.S. (1749-1827). Mathematician. 1, 2, 19, 31, 35
Lardner, Dionysius (1793-1859). First professor of natural
philosophy at London University (i.e. University
College London), a Fellow of the Royal Society and
many other learned bodies. 36
Leibniz, G.W. ( 1 646-1 716). Philosopher and mathematician. 41,68
Legendre, A.M. (1752-1823). Mathematician. 58
Libri, Guglielmo (1803-69), Libri was born in Florence,
moved to France where he became naturalized in 1833.
He was Inspecteur des bibliotheques de France. 28, 63
Lloyd, B.C. (1808-72). Provost of Trinity College, Dublin,
1831-7. 34
Logan, H.F.C. (fl. 1830-50). Professor at the Catholic
College at Oscott, and a friend of De Morgan. 13
Luby, Thomas (1800-70). Luby occupied various posts at
Trinity College, Dublin, and wrote astronomical
texts. 34
Lumley, Edward (fl. 1 830-40). A bookseller who had a
shop in Chancery Lane. 35
134 BIOGRAPHICAL NOTES
Lycurgus (9th century BC). The traditional law-giver of Sparta. 80
MacHale, J. (1791-1881). Archbishop of Tuam from
1834 to 1881. 24
MacMillan, Daniel (1813-57) and Alexander (1818-96).
The founders of the firm of publishers. 63
Mansel, H.L. (1820—71). Philosopher, sometime Reader
in Theology, Oxford. 60, 64, 74, 77
Maynard, S. (1740-1862). A mathematical bookseller who
wrote and edited mathematical texts. 64
Mulcahy, John (d. 1853). Professor of Mathematics,
Queen's College, Galway, 1 849-53. 48
Murphy, Robert (1806—43). Mathematician, sometime
Fellow and Dean of Caius College, Cambridge. 48
Napier, John (1550-1617). Inventor of logarithms. 89
Newman, F.W. (1805—97). Professor of classics at
Manchester University and of Roman Literature
at University College London. 64
Newton, Isaac (1642—1727). Physicist and mathematician. 41, 42
O'Brien, J.T. (1792-1874). Fellow of Trinity College,
Dublin 1820-36. 39
O'Higgins, William (d. 1853). Bishop of Ardagh 1829-53. 24
Pacius (G. Pace) (1 550-1 633). Pacius wrote on philosophy. 70
Paul of Tarsus (First century). The Apostle Paul. 70, 72
Peyrard, Francis (1760—1822). A scholar who edited
Euclid's and Archimedes works. 69
Pfaff, J.F. (1765-1825). Writer on differential equations;
he was a close friend of Gauss. t 84
Poisson, S.D. (1781-1840). Poisson worked at the Ecole
Poly technique on mathematical physics. 30, 35
Powell, Baden (1796-1860). Savilian Professor of
Geometry at Oxford. 67
Puissant, Louis (1769-1843). Author of books on geodesy
and astronomy. 58
Quilp, Daniel. A character in Dicken's The Old Curiosity
Shop. 38
Renan, Ernest (1823—62). Renan wrote on biblical, Jewish,
and linguistic subjects as well as the well-known
La Vie de Jesus. 79
Ryall, Dr John (d. 1875). Professor of Greek at Queen's
College, Cork. 26
Sadleir, F. (1774-1861). Fellow and Professor of
Mathematics at Trinity College, Dublin. 39
Saul of Tarsus (First century). The Apostle Paul. 70
Shakespeare, William (1 564-1616). Dramatist and poet. 48
Shylock. Character in Shakespeare's The Merchant of
Venice. 70
Solly, T. (1813-75). Professor of English at Berlin
University from 1 843 to 1 875. 13
Stevens. Not identified (possibly Henry Stevens 1819-86,
bibliophile, bibliographer, and bookseller, an
American who settled in London in 1 845). 56
BIOGRAPHICAL NOTES 135
Sylvester, James, J. (1814-97). Algebraist who contributed
to the theory of quadratic forms and invariants. 30
Taylor, Richard (1781-1858). Editor of the Philosophical
Magazine from 1822; a fellow of the Linnean,
Astronomical, and other learned societies. 32, 38, 39, 41
Thomson, William (1819-90). Sometime Tutor, The
Queen's College, Oxford; Archbishop of York,
1862-90. 13
Todhunter, Isaac (1820-84). Fellow of St John's College,
Cambridge, and the author of many textbooks. 65, 83
Veitch, John (1829-94). Professor of Logic at St Andrews
and Glasgow Universities. 77
Venetus, Paulus (Niccoletti, Paolo) (d. 1429). Philosophical
writer. 70
Wagner, Gabriel (fl. 1 700). A controversialist who opposed
late seventeenth century scholasticism ; he wrote
under the pseudonym Realis de Vienna. 78
Walker, John (fl. 1790-1830). Fellow of Trinity College,
Dublin. 64
Wallis, John (1616-1703). Mathematician, Savilian
Professor of Geometry, Oxford, 1649-1703. 70
Walsh, John (1786-1847). An eccentric who lived in Cork. 27-31, 39-41
83
Wedgwood, Hensleigh (1803-91). A barrister and
magistrate who who wrote on philology. 64
Whewell, William (1794-1866). Master of Trinity College,
Cambridge, 1841-66 and Knightbridge Professor of
Moral Philosophy, 1838-55. * 3
Young, J.R. (1799-1885). Professor of Mathematics in the
Belfast Institution, 1833 to 1849, and author of
many textbooks. 2 1 , 64
Zunz, Leopold ( 1 794- 1886). Author. 7 1
BIBLIOGRAPHY
The bibliography is arranged in two main parts, first manuscripts, second printed
works.
Manuscripts
This part of the bibliography is arranged in two sections:
(a) The letters of Boole and De Morgan
(b) Other manuscripts.
(a) The letters of Boole and De Morgan
With five exceptions - Letters 10, 20, 73, 78, 79 - all the letters are in the
library of University College, London. They are catalogued under the number
MS Add 97.
The Letters 10, 20, 73, 78, and 79 are to be found in the library of London
University. Letter 20 is pasted in a bound collection of pamphlets (once part of
De Morgan's library), and is catalogued under G. Boole, On a General Method in
Analysis, 1844, L° [B.P. 21]. Letters 10, 73, 78, and 79, all from Boole to
De Morgan, are designated MS 775/370/5, items 1 to 4. MS 775/370 is a col-
lection of letters to and from De Morgan.
No.
Writer
Date
No.
Writer
Date
1
DeM
29 Dec. 1842
22
B
3 Sept.
2
B
19 June 1843
23
DeM
4 Sept.
3
DeM
24 Nov.
24
B
8 Nov.
4
B
8 Dec.
25
DeM
8 June 1850
5
DeM
1 1 Dec.
26
B
17 Oct.
6
B
28 June 1844
27
B
31 Mar. 1851
7
B
15 Jan. 1845
28
B
22 Apr.
8
DeM
6 Feb.
29
B
6 May
9
B
24 Feb.
30
B
25 May
10
B
8 Jan. 1847
31
B
24 June
11
DeM
31 May
32
B
16 July
12
DeM
28 Nov.
33
B
24 July
13
DeM
29 Nov.
34
B
29 July
14
B
24 Aug. 1848
35
B
4 Aug.
15
B
8 Dec.
36
B
1 1 Aug.
16
DeM
3 Apr. 1849
37
B
25 Aug.
17
B
12 Apr.
38
B
9 Sept.
18
B
21 Apr.
39
B
10 Sept.
19
DeM
10 June
40
B
17 Nov.
20
B
13 Aug.
41
B
28 Nov.
21
DeM
14 Aug.
42
B
28 June 1852
BIBLIOGRAPHY
137
No.
Writer
Date
No.
Writer
Date
43
B
12 July
67
B
13 Nov.
44
B
23 July
68
B
7 Feb. 1861
45
B
27 Sept.
69
B
17 May
46
B
8 Oct.
70
DeM
16 Oct.
47
B
8 Dec.
71
B
4 Nov.
48
B
7 Feb. 1854
72
B
21 Nov.
49
B
15 Feb.
73
B
7 Jan. 1862
50
B
23 Feb.
74
DeM
1 Feb.
51
B
30 May
75
B
12 Feb.
52
B
n.d.
76
B
21 Apr.
53
B
3 Jan. 1855
77
DeM
20 Sept.
54
B
3 Feb.
78
B
4 Nov.
55
B
21 Feb.
79
B
6 Nov.
56
DeM
4 Jan. 1856
80
DeM
7 Nov.
57
B
8 Jan.
81
B
10 Nov.
58
DeM
13 Jan.
82
DeM
12 Nov
59
B
23 Feb.
83
B
3 Jan. 1863
60
B
21 Mar. 1859
84
B
7 Jan.
61
B
9 June
85
DeM
7 Feb.
62
B
15 Sept.
86
DeM
5 July
63
DeM
10 June 1860
87
DeM
8 Aug.
64
DeM
13 July
88
DeM
25 Apr. 1864
65
B
17 July
89
B
3 May
66
B
18 Oct.
90
DeM
n.d.
(b) Other manuscripts
De Morgan, A. (1860). [Autobiographical Note] British Library Add MS 28, 509
f.421.
(1864). Draft letter to H.M. Government. University College London
Library. MS Add 97/1.
(1867). Note on Professor Boole's Papers. Library of the Royal Society.
MS M.M. 16.34.
Boole, G. (1847). Draft letter to A. De Morgan, 8 Jan. 1847. Royal Society
Library, Boole papers W8.
Roy. Soc. [part of A97 of Boole papers]. Library of the Royal Society.
Printed works
This part of the bibliography is arranged in three sections:
(a) Boole
(b) De Morgan
(c) Other authors.
For authors other than G. Boole and A. De Morgan titles of periodicals, etc.
are given in full. For G. Boole and A. De Morgan titles of periodicals have been
indicated by initials: a list of these abbreviations is given on page 138.
Collective works (encyclopaedias, etc.) with no identifying author's name are
assigned a brief title in place of an author: thus The English Cyclopaedia is
designed Eng. Cyc. 1854 and will be found in the listing under 'Eng. Cyc.'.
[R] preceding an entry should be read as 'Review of. [Rs] following an entry
means 'Review of books'. 'SDUK' indicates that a book was published under the
auspices of the Society for the Diffusion of Useful Knowledge.
The parts of the Cambridge MathematicalJournal and the Transactions of the
138 BIBLIOGRAPHY
Cambridge Philosophical Society were issued earlier than the volumes; I have
assigned papers in these periodicals to the date of the part (which is given after
the page reference) rather than to that of the volume.
Abbreviations of titles of periodicals used in the G. Boole and A. De Morgan
bibliographies
AM Assurance Magazine
ASM Astronomical Society, Memoirs (*)
ASMN Astronomical Society, Monthly Notices (*)
Ath Athenaeum (*)
BA British Association for the Advancement of Science, Report
BM Bentley's Miscellany (*)
CA Companion to the Almanac
CDMJ Cambridge and Dublin Mathematical Journal
CJM Crelle's Journal der Mathematik
CMJ Cambridge Mathematical Journal
CPST Cambridge Philosophical Society, Transactions
CSEP Central Society of Education, Publications
DR Dublin Review
GM Gentlemen's Magazine (*)
IAJ Institute of Actuaries, Journal
LJM Liouville's Journal de Mathematique
LMSP London Mathematical Society, Proceedings
Ma Mathematician, The
MM Mechanics Magazine
NBR North British Review
NR National Review
NQ Notes and Queries (*)
PM Philosophical Magazine
PSP Philological Society, Proceedings (*)
PST Philological Society, Transactions (*)
QJE Quarterly Journal of Education (*)
QJM Quarterly Journal of Pure and Applied Mathematics
RIAP Royal Irish Academy, Proceedings
RIAT Royal Irish Academy, Transactions
RSET Royal Society of Edinburgh, Transactions
RSP Royal Society, Proceedings
RSPT Royal Society, Philosophical Transactions
SPAB St Petersburg Academy, Bulletin.
* May contain other items by De Morgan.
(a) G. Boole
I have tried to make the bibliography of Boole's work complete; I should add
that substantial portions of letters of Boole are included in Cayley (1362a)
(= Cayley 1889, vol. 5, 80-84) and in Jourdain (1913).
1835 An address on the Genius and Discoveries of Sir Isaac Newton. Lincoln
1835.
1840a Researches on the theory of analytical transformations. CMJ 2 (1841)
64-73 (Feb. 1840).
b On certain theorems in the calculus of variations. CMJ 2(1 84 1)97— 1 02
(May 1840).
c On the integration of linear differential equations with constant coef-
ficients. CMJ 2 (1841) 1 14-19 (May 1840).
PRINTED WORKS 139
d Analytical Geometry. CMJ 2 (1841) 178-88 (Nov. 1849).
1841 Exposition of a general theory of linear transformations. CMJ 3 (1842)
1-20 (Nov. 1841), 100-1 19 (May 1842).
1843a On the transformation of definite integrals. CMJ 3 (1843) 216-24
(Feb. 1843).
b Remarks on a theorem of M. Catalan. CMJ 3 (1843) 277 -83 (May
1843).
c On the transformation of multiple integrals. CMJ 4 (1845) 20-28
(Nov. 1843).
1844a On a general method in analysis. RSPT 134 (1844) 225-82.
b On the inverse calculus of definite integrals. CMJ 4 (1845) 82-7 (Feb.
1844).
c Notes on linear transformations. CMJ 4 (1845) 167-71 (Nov. 1844).
1845a On the theory of developments, Part 1. CMJ 4 (1845) 214-23 (Feb.
1845).
b On the equation of Laplace's functions. BA (1845) Part 2, 2.
1846 On the equation of Laplace's functions. CDMJ 1 (1846) 10-22.
1 847a The Mathematical Analysis of Logic. Cambridge 1 847 (reprinted Oxford
1948, 1951).
b The Right Use of Leisure. London 1847.
c On the attraction of a solid of revolution on an external point. CDMJ 2
(1847) 1-7.
d On a certain symbolical equation. CDMJ 2 (1847) 7—12.
e Remarks on the Rev. B. Bronwin's method for differential equations.
PM (3)30(1847)6-8.
/ Note on a class of differential equations. PM(3) 30 ( 1 847) 96-7.
1848a Remarks on a paper by the Rev. B. Bronwin. On the solution of a par-
ticular differential equation. PM{3) 32 (1848) 413-15.
b Remarks on a paper by the Rev. B. Bronwin. On the solution of a par-
ticular differential equation. PM(3) 33 (1848) 211.
c Note on quaternions. PM(3) 33 ( 1 848) 278-80.
d Theoreme general concernant Fintegration definie. LJM 13 (1848)
111-12.
e On the analysis of discontinuous functions. RIAT 21 (1848) 124—39.
/ On a certain multiple definite integral. RIAT 21 (1848) 140-49.
g On a general transformation of any quantitative function. CDMJ 3
(1848) 112-16.
h On the calculus of logic. CDMJ 3 (1848) 183-98.
i Mr Boole's theory of the mathematical basis of logic. MM 49 (1848)
254-5.
1849 On a general theorem of definite integration. CDMJ 4 (1849) 14-20.
1851a The Claims of Science. London 1851.
b On the theory of linear transformations. CDMJ 6 (1851) 87-106.
c On the reduction of the general equation of the nth degree. CDMJ 6
(1851) 106-13.
d Letter to the Editor. CDMJ 6 (1851) 284-5.
e Proposed question on the theory of probabilities. CDMJ 6 (1851) 286.
/ On the theory of probabilities and in particular on Mitchell's Problem
of the distribution of the fixed stars. PM(4) 1(1851)521 -30.
g Further observations on the theory of probabilities. PM(4) 2 (1851)
96-101.
h An account of the late John Walsh of Cork. PM(4) 2 (1851) 348-58.
140 BIBLIOGRAPHY
1852 On reciprocal methods in the differential calculus. CDMJ 7 (1852)
156-66.
1853 On reciprocal methods in the differential calculus, continued. CDMJ %
(1853) 1-24.
1854a An Investigation into the Laws of Thought. London, 1854; for another
edition (reprinted 1958 New York) see 1916.
b Solution of a question in the theory of probabilities. PM{4) 7 (1854)
29-32.
c Reply to some observation of Mr. Wilbraham on the theory of chances
developed in Prof. Boole's Laws of Thought. PM(4) 8 (1854) 87-91.
d On the conditions by which the solutions of questions in the theory of
probabilities are limited. PM(4) 8 (1854) 91-8.
e Further observations relating to the theory of probabilities in reply to
Mr. Wilbraham. PM(4) 8 (1854) 175-6.
/ On a general method in the theory of probabilities. PM(4) 8 (1854)
431-44.
1855c The Social Aspects of Intellectual Culture. Cork, 1855.
b On certain propositions in algebra connected with the theory of prob-
abilities. PM(4) 9 (1855) 165-79.
1856 On the solution of the equation of continuity of an Incompressible
Fluid [letter to C. Graves, 5 May 1856]. RIAP 6 (1853-57) 375-85.
1857c On the comparison of transcendents with certain applications to the
theory of definite integrals. RSPT 147 (1857) 754-804.
b On the application of the theory of probabilities to the question of the
combination of testimonies or judgements. RSET 21 (1857) 597-652.
1859 Treatise on Differential Equations. London, 1859. 2nd edition 1865,
3rd 1872, 4th 1877, and a number of later reprints.
1860 Treatise on the Calculus of Finite Differences. London, 1869, 2nd
edition 1872, 3rd 1880, and a number of later reprints.
1862a On the theory of probabilities. RSPT 152 (1862) 225-52.
b On simultaneous differential equations of the first order in which the
number of variables exceeds by more than one the number of the
equations. RSPT 152 (1862) 437-54.
c On the integration of simultaneous differential equations. RSP 12
(1862-3) 13-16.
d On the theory of probabilities. RSP 12(1 862-3) 1 79-84.
e On the differential equations of dynamics. RSP 12 (1862-3) 420-24
/ On a question in the theory of probabilities. PM(4) 24 ( 1 862) 80.
g Considerations sur la recherche des integrates premieres des equations
differentielles partielles du seconde ordre. SPAB IV (1862) col.
198-215.
h On simultaneous differential equations in which the number of vari-
ables exceeds by more than unity the number of equations. RSP 12
(1862-3) 184.
i Supplement to a paper 'On the differential equations of dynamics'.
RSP 12 (1862-3) 481.
1 863a On the differential equations of dynamics. A sequel to a paper on simul-
taneous differential equations. RSPT 153 (1863) 485-501.
b Uber die partielle Differentialgleichungen zweiter Ordnung Rr + Ss +
Tt + U(s 2 -rt) = V. CJM 61 (1863) 309-33.
1 864a On the differential equations which determine the form of the roots of
an algebraic equation. RSPT 154 (1864) 733-55.
PRINTED WORKS 141
b On the differential equations which determine the form of the roots of
an algebraic equation. RSP 13 (1864) 245-6.
1865 Treatise on Differential Equations. Supplementary Volume, editor I.
Todhunder. Cambridge 1865.
1868 On propositions numerically definite (read posthumously by A. De
Morgan, March 1868). CPST 11 (1871) 396-411.
1916 Collected Logical Works, vol. II. Chicago and London 1916. Edited by
P.E.B. Jourdain. (Note that no vol. I was issued: vol. II was reprinted
in 1940 and 1952. The bulk of the volume is a reprint of 1854a.)
1952 Studies in Logic and Probability. London and La Salle 1952. Edited by
R. Rhees. (This volume fills the gap caused by the non-appearance of
vol. I of Collected Logical Works. Some editions are titled: Collected
Logical Works Vol. I, Studies, etc. It prints for the first time some
Boole manuscripts now in the Library of the Royal Society, together
with the following items: 1847a, 1848/*, 1868, 1851a, 1851/, 1851g,
1854&, 1845c, 1854J, 1854e, 1845/, 18576, 1862a.)
(b) A. De Morgan
De Morgan wrote prolifically for a wide range of periodicals; the periodicals
marked (*) in the list on page 138 may contain other items by De Morgan — this
is certainly the case for Notes and Queries and the Athenaeum. For the peri-
odicals not so marked, I hope the bibliography is complete. The only items I
have deliberately omitted are certain reviews of elementary textbooks in the
Quarterly Journal of Education and the brief reports of the papers read before
the Cambridge Philosophical Society in the Proceedings: these merely summarize
the papers in the Transactions. In addition I have not attempted to include a
number of prefaces, introductions, and indexes he wrote in books by other
authors: see S.E. De Morgan 1882, 415, for further information on such items.
1828 The Elements of Algebra. Translated by A. De Morgan from the first
three chapters of the Algebra of M. Bourdon. London 1828.
1830a The Elements of Arithmetic. London 1830. 2nd edition 1832, 3rd
1835, 4th 1840, 5th 1846, 6th 1876, and many reprints).
b Remarks on Elementary Education in Science. London 1830.
c On the general equations of curves of the second degree. CPST 4 ( 1 833)
71-8 (Nov. 1830).
1831a The Study and Difficulties of Mathematics. SDUK. London 1831
(reprinted 1832, 1836, 1840, 1 847 ; Chicago edition 1898, 1902, 1910;
La Salle edition 1943).
b On life assurance. CA 1831 86-105.
c On mathematical instruction. QJE 1 (1831) 264-79.
d Polytechnic School of Paris. QJE 1 (1831) 57-86.
1832a Elementary Illustrations of the Differential and Integral Calculus.
SDUK. London 1832 (2nd edition 1842; Chicago editions 1899, 1909;
La Salle edition 1943).
b On the general equation of the surfaces of the second degree. CPST 5
(1835) 77-94 (Nov. 1832).
c On eclipses. CA 1832 5-12.
d Study of natural philosophy. QJE 3 (1 832) 60-73.
e On some methods employed for the instruction of the deaf and dumb.
QJE 3 (1832) 203-19.
/ State of mathematical and physical sciences in the University of Oxford.
QJE 4 (1832) 191-208.
142 BIBLIOGRAPHY
1833a A new method of reducing the apparent distance of the moon. ASM 5
(1833)245-52.
b On comets. CA 1833 5-15.
c On teaching arithmetic. QJE 5(1833)1-16.
d On the method of teaching fractional arithmetic. QJE 5 (1 833) 210—22.
e On the method of teaching the elements of geometry. Pt. 1. QJE 6
(1833)35-49.
/ On the method of teaching geometry. Pt. 2. QJE 6 (1833) 237-51.
1834a The Elements of Spherical Trigonometry. SDUK. London 1834.
b On the moon's orbit. CA 1834 5-23.
c On the notation of the differential calculus adopted in some works
lately published at Cambridge. QJE 8 (1834) 100-1 10.
d [R] Airy's Gravitation. QJE 8 (1834) 316-25.
1835a The Elements of Algebra preliminary to the Differential Calculus.
London 1835 (2nd edition 1837).
b Examples of the Processes of Arithmetic and Algebra. SDUK. London
1835.
c On Taylor's theorem. PM(3) 7 (1835) 188-92.
d Halley's comet. CA 1835 5-15.
e [R] Peacock's Treatise on Algebra, e/fi" 9 (1835) 91-110, 293-311.
/ Ecole polytechnique. QJE 10 (1 835) 330-40.
1836a The Connexion of Number and Magnitude. London 1836.
b Examples of the processes of arithmetic and algebra. London 1836
(reprinted 1847).
c An Explanation of the Gnomic projection of the sphere. SDUK. London
1836.
d A Treatise on the Calculus of Functions [contribution to Encyclopaedia
Metropolitan] London 1836, 305-92.
e Old arguments against the motion of the Earth. CA 1836 5-19.
/ On the relative signs of coordinates. PM(3) 9 (1836) 249-54.
g A sketch of a method of introducing discontinuous constants. CPST 6
(1838) 185-93 (May 1836).
h [Contributions: Bradley, Delambre, Descartes, Dollond, Euler, Halley,
Harrison, W. Herschel, Lagrange, Laplace, Leibniz and Maskelyne in
The Gallery of Portraits: with Memoirs, London 1836.]
1837a The Elements of Trigonometry and Trigonometrical Analysis prelimin-
ary to the Differential Calculus. London 1837.
b Thoughts suggested by the Establishment of the University of London.
London 1837.
c Theory of Probabilities [contribution to Encyclopaedia Metropolitan] .
London 1837.
d On a question in the theory of probabilities. CPST 6 (1837) 423-30
(Feb. 1837).
e Notices of English mathematical and astronomical writers between the
Norman conquest and the year 1600. CA (1837) 21-44.
/ [R] Theoreme Analytique des Probabilites. Par M. Le Marquis de
Laplace. 3eme edn. 1820.2^ 3 (Apr. 1837) 338-54.
g [R] Theoreme Analytique des Probabilites. Par M. Le Marquis de
Laplace. 3eme edn. 1820. DR 3 (July 1837) 237-48.
h The Mathematics; their Value in education. CSEP 1 (1837) 114-44.
1838a An Essay on Probabilities. London 1838 (2nd edtion 1841, 3rd 1849).
b On the solid polyhedron. PM(3) 12 (1939) 323-4.
PRINTED WORKS 143
c On Cavendish's experiment. CA 1838 26-42.
d Professional mathematics. CSEP 2 (1838) 132-47.
1 839a First Notions of Logic. London 1 839 (2nd edition 1 840).
b On the foundation of algebra. CPST 7 (1841) 173-187 (Dec. 1839.)
c On the rule for finding the value of an annuity for three lives. PM(3) 15
(1839)337-9.
d Notices of the Progress of the problem of evolution. CA 1839 33—51.
1 840a Description of a calculating machine. RSP 4 ( 1 840) 243-4.
b Description of a calculating machine invented by Mr. T. Fowler. PM{3)
17(1840)385-6.
c On the perspective of the coordinate planes. CMJ 2 (1841) 92—3 (Feb.
1840).
d On the calculation of single life contingencies. CA 1840 5—23.
e The necessity of legislation for life assurance. DR (Aug. 1840) 49—88.
1841a On the foundation of algebra II. CPST1 (1842) 287-300 (Nov. 1841).
b Suggestion on Barrett's method of computing the values of life con-
tingencies. PM(3) 18 (1841) 268-70.
c On Fernel's measure of a degree. PM(3) 19 (1841) 445-447.
d On a simple property of the conic sections. CMJ 2 (1841) 202—3 (Feb.
1841).
e Remarks on the binomial theorem. CMJ 3 (1843) 61-2 (Nov. 1841).
/ On the use of small tables of logarithms in commercial calculations, and
on the practicability of a decimal coinage. CA 1841 5—20.
g [R] Jones' The values of annuities and reversionary payments. DR 11
(Aug. 1841) 104-133.
h [R] Peyrard's Elements of Euclid. DR 11 (Nov. 1841) 330-55.
i [R] Report on the commissioners appointed to consider the steps to be
taken for restoration of the standards of weights and measures. DR 12
(Nov. 1842)466-93.
/ A short way of reducing the square root of a number to a continued
fraction. CMJ 2 239-40 (Feb. 1841).
1 842a The Differential and Integral Calculus. SDUK. London 1 842.
b Additional note on the history of Fernel's measure of a degree. PM{3)
20(1842) 116-17.
c On Fernel's measure of a degree ; in reply to Mr Galloway's remarks.
PM{3) 20 (1842) 230-33.
d On Fernel's measure of a degree. PM(3) 20 (1842) 408-1 1.
e On Leonardo da Vinci's use of + and — PM(3) 20 (1842) 135-7.
/ On life contingencies, No. 2. CA 1 842 1 -1 8.
g Science and rank. DR 1 3 (Nov. 1 842) 4 1 3 -48.
1843a On the invention of the circular parts. PM 22 (1843) 350-53.
b On the almost total disappearance of the earliest trigonometric canon.
ASMN6 (1843-51) 221-8.
c On the foundation of algebra III. CPST 8 ( 1 844) 139-42 (Nov. 1 843).
d On Torporley's anticipation of part of Napier's rule. PM(3) 22 (1843)
350-53.
e References for the history of the mathematical sciences. CA (1843)
40-65.
1844a On the reduction of a continued fraction to a series. PM(3) 24 (1844)
15-17.
b On the foundation of algebra IV. CPST 8 (1847) 241-54 (Oct. 1844).
144 BIBLIOGRAPHY
c On divergent series and various points of analysis connected with them.
CPST 8 (1849) 182-203 (Mar. 1844).
d On the equation (£> + a) n y = X. CMJ 4 (1845) 60-62 (Feb. 1844).
e Addendum to 'On the equation (D + a) n y =X\ CMJ 4 ( 1 845) 96 (May
1844).
/ On the law existing in the successive approximations to a continued
fraction. CMJ 4 (1845) 97-9 (May 1844).
g On arithmetical computation. CA 1 844 1—21.
1845c The Globes Celestial and Terrestrial. SDUK. London. (2nd edition 1847,
3rd 1854).
b On the ecclesiastical calendar - easter. CA (1845) 1-36.
c Baily's repetition of the Cavendish experiment on the mean density of
the earth [Rs]. DR 18 (Mar. 1845) 75-1 12.
d Speculators and Speculation [Rs]. DR 19 (Sept. 1845) 99-129.
e Book-keeping [Rs]. DR 19 (Dec. 1845) 433-53.
1846c On Arbogast's Formula of expansion. CDMJ 1 (1846) 238-55.
b On a point connected with the dispute between Keill and Leibnitz
about the invention of fluxions. RSPT 136 (1846) 107-9.
c On the structure of the syllogism [sometimes referred to as 'On the
Syllogism I']. CPST 8 (1849) 379-408 (Nov. 1846).
d Newton. Contribution to vol. xi of the Cabinet Portrait Gallery of
British Worthies. London. 78—117.
e Derivation of the word theodolite. PM(3) 28 (1846) 287-9.
/ Derivation of tangent and secant. PM{3) 28 (1846) 382-7.
g On the earliest printed almanacs. CA 1 846 1—31.
h Mathematical bibliography [Rs]. DR 21 (Sept. 1846) 1-37.
1847c Arithmetical Books from the Invention of Printing to the present Time.
London.
b Formal Logic. London (reprinted 1926, Editor A.E. Taylor).
c Statement in Answer to an Assertion made by Sir W. Hamilton. London
1847.
d Recurrences of eclipses and full moon. CA 1847 53—5.
e On helps to calculation [R 'Tables' in Penny Cyclopaedia, Supplement].
DR 22 (Mar. 1847)74-92.
1848c Methods of integrating partial differential equations. CPST 8 (1849)
606-13 (June 1848).
b Suggestions on the integration of rational fractions. CDMJ 3 (1848)
238-42.
c Account of the speculations of Thos. Wright of Durham. PM{3) 32
(1848)241-52.
d On the additions made to the second edition of the Commercium Epi-
stolicum. PM{3) 32 (1848) 446-56.
e On a property of the hyperbola. PM(3) 33 (1848) 546-8.
/ On decimal coinage. CA 1848 5—21.
g Remark on the general equation of the second degree. Ma 3 (1848)
154-5.
1 849c Trigonometry and Double Algebra. London.
b On a point in the solution of linear differential equations. CDMJ 4
(1849) 137-9.
c On anharmonic ratio. PM(3) 35 (1849) 165-71.
PRINTED WORKS 145
d Short supplementary remarks on the first six books of Euclid's Ele-
ments. CA (1849) 5-20.
e Organised method of making the reduction required in the integration
of rational fractions. Ma 3 (1849) 242-6.
/ Remarks on Homers method of solving equations. Ma 3 ( 1 849) 289-9 1 .
1850a On the symbols of logic, the theory of the syllogism, and in particular
of the copula [sometimes cited as 'On the Syllogism II']. CPST 9
(1856) 79-127 (Feb. 1850).
b Extension of the word 'area'. CDMJ 5 ( 1 850) 1 39-42.
c Supplement to the remark on the general equations of the second
degree. Ma 3 (1850) supplement 4-5.
d On ancient and modern usage in reckoning. CA (1850) 5—33.
1851a The Book of Almanacs. London. (2nd edition 1871, 3rd 1907).
b On some points of the integral calculus. CPST 9 (1856) 107-38 (Feb.
1851).
c Application of combinations to the explanation of Arbogast's method.
CDMJ 6 (1851) 35-7.
d On the mode of using the signs + and — in plane geometry. CDMJ 6
(1851) 156-60.
e On the connexion of involute and evolute in space. CDMJ 6 (1851)
267-74.
/ On some points in the history of arithmetic. CA (1851) 5-18.
g On the equivalence of compound interest with simple interest paid
when due. AM 1 (1851) 335-6.
1852a On the partial differential equations of the first order. CDMJ 7 (1852)
28-35.
b On the signs + and — in geometry and on the interpretation of the
equation of a curve. CDMJ 7(1852) 242-5 1 .
c On indirect demonstrations. PM(4) 3 (1 852) 435-8.
d On the authorship of the 'Account of the Commercium Epistolicum'
published in the Philosophical Transactions. PM(4) 3 (1852) 440-44.
e On the early history of infinitesimals in England. PM(4) 4 (1852)
321-30.
/ A short account of some recent discoveries in England and Germany
relative to the controversy on the invention of fluxions. CA (1852)
5-20.
g The case of M. Libri.fiM 32 (1852) 107-15.
h On a method of checking annuity tables at different rates of interest by
help of one another. AM 2(1852) 380-9 1 .
1 8 5 3a Mathematical notes. CDMJ 8(1853)93-4.
b On the difficulty of correct descriptions of books. CA (1853) 5-19.
c Some suggestions in logical phraseology. PSP Feb. 1853.
1854a On some points in the theory of differential equations. CPST 9 (1866)
513-44 (Mar. 1854).
b On a decimal coinage. CA (1854) 5-16.
c Account of a correspondence between Mr George Barrett and Mr Francis
Baily.^M4(1854) 185-99.
d On the demonstration of formulae connected with interest and annuities.
AM 4 (1854) 277-82.
1855a Summary of decimal coinage questions. London.
b On the singular points of curves and on Newton's theory of coordinated
exponents. CPST 9 (1856) 608-27 (May 1855).
146 BIBLIOGRAPHY
c [R] Brewster's Memoirs of the Life of Sir I. Newton. NBR 23 (1855)
308-38.
d The progress of the doctrine of the earth's motion between the times of
Copernicus and Galileo, being notes on the ante-Galileo Copernicans.
CA (1855) 5-24.
e On some questions of combinations. AM 5 (1855) 93—9.
1856a On the question What is the solution of a differential equation. CPST
10 (1864) 21-6 (Apr. 1856).
b Notes on the history of the english coinage. CA (1 856) 1 -20.
1857a On the beats of imperfect consonances. CPST 10 (1864) 129-45 (Nov.
1857).
b A proof of the existence of a root in every algebraic equation. CPST 19
(1864) 261-70 (Dec. 1857).
c On the dimensions of the roots of equations. QJM 1 (1857) 1-3, 80.
d Historical note on the theorem respecting the dimensions of roots. QJM
1(1857)232-5.
e Note on Euclid I, 47. QJM 1 (1857) 236-7.
/ Notes on the state of the decimal coinage question. CA (1857) 5-19.
1858a On the syllogism III. CPST 10 (1864) 173-320 (Feb. 1858).
b On the integrating factor of Pdx + Qdy + Rdz. QJM 2 (1858) 323-6.
c On the classification of polygons of a given number of sides. QJM 2
(1858)340-41.
1859a On the general principles of which the composition or aggregation of
forces is a consequence. CPST 10 (1864) 290-304 (Mar. 1859).
b On the word apiBfxoq.PST (1859) 9-14.
1 860a Syllabus of a Proposed System of Logic. London.
b On the syllogism IV. CPST 10 (1864) 331-58 (Apr. 1860).
c On the syllogism of transposed quantity. CPST 10 (1864) 355*-8*
(Apr. 1860).
d On the determination of the rate of interest of an annuity. AM 8 ( 1 860)
61-7.
e On a property of Mr Gompertz's law of mortality. AM 8 ( 1 860) 181-4.
1861a On the theory of errors of observation. CPST 10 (1864) 409-27 (Nov.
1861).
b Logic. Contribution to the English Cyclopaedia, vol. 5, 1861 cols.
340-54.
c [Mathematical] Tables. Contribution to the English Cyclopaedia, vol. 7,
1861, cols. 976-1016.
d On the unfair suppression of the acknowledgement to the writings of
Mr Benjamin Gompertz. AM 8 (1861) 86-9.
e Newton's table of leases. AM 9 (1861) 185-7.
/ On Gompertz's law of mortality. AM 9 (1861) 214-15.
1863a On the syllogism V. CPST 10 (1864) 428-88 (May 1863).
b Note [on Woolhouse's recent paper on annuity on three lives]. AM 10
(1863)237-8.
c On the rejection of the fractions of a pound in extensive valuations. AM
10(1863)247-51.
d A query about interest accounts. AM 10(1 863) 381-2.
e On the forms under which Barrett's method is presented, and on changes
of words and symbols. AM 10 (1863) 301-12.
1864a On infinity and on the sign of equality. CPST 11 (1866) 145-89 (Mar.
1864).
PRINTED WORKS 147
b A theorem relating to neutral series. CPST 11 (1866) 190-202 (May
1864).
c On the early history of the signs + and -. CPST 11 (1866) 203-12
(Nov. 1864).
d Rules to be observed in converting the parts of one pound into decimals.
AM 11 (1864) 53-4.
1865a A theorem relating to neutral series. PM(4) 30 (1865) 372-3.
b On infinity and on the sign of equality. PM(4) 30 ( 1 865) 373-4.
c On the early history of the signs + and — . PM(4) 30 ( 1 865) 376.
1866a A proof that every function has a root. LMSP 1 (1866) 13-14.
b The portrait of Copernicus. GM 168 June 1866 804-8.
c On the root of any function and on neutral series No. 2. CPST 11
(1869) 239-66 (May 1869).
d On the calculation of single life contingencies. AM 12 ( 1 866) 328-49.
e On a problem in annuities, and on Arbogast's method of development.
AM 12 (1866) 206-12.
/ On the summation of divergent series. AM 12 (1866) 245—52.
g Speech at the first meeting of [the London Mathematical] Society.
LMSP I (1866) 1-9.
1867 On the calculation of single life contingencies. AM 13 (1867) 129-49.
1868a On the conic octagram. LMSP 2 (1868) 26-9.
b On general numerical solutions. LMSP 2 (1868) 84—5.
c Note on 'A theorem relating to neutral series' in vol. 11, pt. 2. CPST 1 1
(1869) 447-60 (Oct. 1868).
1 869a Fourier's statistical tables. IAJ 14 ( 1 869) 89-90.
b On the final law of the sums of drawings. IAJ 14 (1869) 175-82.
1870 Remark on [Woolhouse's paper 'On general numerical solutions']. IAJ
15(1870)327-8.
1872 A Budget of Paradoxes. Edited by S.E. De Morgan. London (2nd
edition 1915, ed. D.E. Smith, reprinted 1954).
1885 Newton, his Friend, and his Niece. Edited S.E. De Morgan and A.C.
Ranyard. London.
1914 Essays on the Life and Works of Newton. Notes and appendices by
P.E.B. Jourdain. London [contains 1846d, 1852/, 1855c].
1966 On the Syllogism and Other Logical Writings. Edited with an Intro-
duction by P. Heath. London.
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Airy, G.B. (1896). Autobiography of Sir George Biddell Airy. Editor W. Airy.
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INDEX
A.B. (pseudonym of A. De Morgan) 107,
112,114,116
Academie des Sciences 80
Airy,G.B. 64
America 103
Ampere, A.M. 99
analytical dynamics 65 , 68
Aristotle 34,37,74,87,107
Arnauld.A. 37
Assurance Journal 118
Astronomical Society 4
Athenaeum 4, 40, 41, 84, 87, 93, 100, 103
Australia 56,64,74
Barrow, I. 75,78
Barry, R. 64
Baynes,T.S. 37,100,103
Beddoes, T. 79
Belfast 33,36,64,79
Belfast Institution 33
Bentley 's Miscellany 17,41
Berkeley, G. 107
Bertrand, J.L.F. 77
Billingsley, H. 76
Birmingham 29
Blackrock 5
Boase.H.S. 79,82
Boethius 87
Boole, Alice 74,79
Boole, Charles 2
Boole, George
bibliography 136-41
biography 1-3
Calculus of Logic 32
Claims of Science 51
Laws of Thought 32, 40, 47, 55-7,
62,117,119
Mathematical Analysis of Logic 17,
22,119
On a General Method in Analysis 7
Boole, John 1,29,79
Boole, Mary Ann 2
Boole, Mary Everest 2, 3, 56, 71, 74,
77,92,105
Boole, William 2
British Museum 17,21
Broadstairs 52,53
Burger sdicius 87
Burgersdijck,F.P. 85
CD. (pseudonym of S.E. De Morgan)
107,110,112,114
Cagnoli, A. 70
calculus of operations 14, 15
calculus of variations 2
Cambridge and Dublin Mathematical Jour-
nal 18,42,46
Cambridge Analytical School 3
Cambridge Mathematical Journal 2, 7, 12,
13,15,116
Cambridge Philosophical Society 13,83
Camden Street 36
Carlyle.T. 80,82
Carnarvonshire 112
Cauchy,A.L. 9,20,76,128
Cayley,A. 47,65,67,96,100
Chretien, C.P. 37
Clebsch, R.F.A. 106, 108, 109
Coleridge, S.T. 110
Cologne Regents 85, 87
Columbus, C. 103
continued fractions 8-10,15
Cork 1, 2, 5, 18, 30-3, 36, 48, 55, 61,
66,67,97
Cournot, A.A. 62
Crackanthorpe, R. 85,87
Creswell, Sir C. 85,90
Crimea 65
Crowther, J.G. 4
D'Alembert, J.L. 107
Davies.T.S. 52,70
De Morgan, Augustus
bibliography 136-8,141-7
biography 3, 4
Budget of Paradoxes 40,119
Formal Logic 17,22-4
On the Syllogism 33, 73
On the Syllogism II 15
154
INDEX
On the Syllogism IV 120
On the Syllogism V 84-6, 93
De Morgan, G.C. 118
De Morgan, H.C. 118
De Morgan, S.E. 4, 5, 17, 18, 38, 40,
51,52,62,69,70,77,107
De Vericour, R. 37, 38, 62
Dedekind, R. 47
Dee, J. 74,76
definite integrals 73
definite multiple integrals 12
Delambre, J.B.J. 70
Dickens, C. 51,53
difference equations 7,8
differential equations 1, 2, 8, 9, 11, 40-2,
65, 76, 77, 97, 103, 106, 107, 119,
128
divergent series 4, 15, 111, 112
Doncaster 2
Donkin.W.F. 50,51
Dublin 97,117
Duns Scotus 91
Ecole Polytechnique 3
Edinburgh Review 45 , 47
Edinburgh University 4
Ellis, RX. 12
Ellis, Sir H. 21
England 1,3,38,66,67,91
Erfurt 85
Esculapius 79
Essays and Reviews 80, 82
Euclid 74,81,83
Euler, L. 80
Euripides 34
finite differences 2,11,79,119
Frankfurt 91
Fraser,A.C. 100
French Institute 117
Frend, Sophia Elizabeth 3
Galway 33,64
Garibaldi, G. 91
Gauss, C.F. 65
German 105,109,110
Germany 91
Gill,M.H. 61,62
Gladstone, W.E. 33
Glaisher, J.W.L. 128
Goodacre 21
Graves 26
Graves, C. 37,66
Greek 34,37,65,68,74
Gregory, D.F. 13,42,44
Gregory, O.A. 79
HaU,DrM. 52
Halperin,T. 57
Hamilton, W. 22, 33, 37, 74-8, 84, 87,
92,102
Hamilton, W.R. 10,55,66,97,118
Harley.R. 3
Heath, P. 5
Heaviside,J.W.L. 77
Heald.W. 52
Herschel, J. 64,75,76
Hildebert 56
Hobbes.T. 107
Holyhead 97
Home,D.D. 114
homoeopathy 74, 78
India 67
infinity 107,110
Ingleby,C.M. 87
invariants 2, 65,67
Ireland 32,33,35,61,64,65,67,96
Irish Colleges 36
Isenach 87
Jacobi, C.G.J. 79, 80, 82, 108
Jamieson, R.A. 81,83
Jews 86,90,91
Jodochus Isenach 85
Johnson, S. 107,110
Journal of the Institute of Actuaries 118
Jowett, B. 80,82,92
Kane,R. 18,38,48
Kant, I. 61,107
Keckermann, B. 85 , 87
KeiU,J. 52
Kelland.P. 75,78
Kneale.W. 3
Knight, C. 75
Lacroix, S.F. 2
Lagrange, J. L. 2
Laplace transform 9
Laplace, P.S. 7,8,10,11,35,45,47,66
Lardner, D. 51
Legendre, A.M. 70,80
Leibniz, G.W. 3,52, 54,56, 94, 101 , 107
Libri,G. 40,43,77
Lincoln 2,5,47,52
Lincolnshire 79
linear equations 12
linear partial differential equations 11
linear transformations 2
Liouville 's Journal 18,125
Locke, J. 107
logic 4, 17, 23-9, 30, 55, 58, 73
INDEX
155
London 17,30,36,58,61,79,92,97
London Mathematical Society 4, 118
London University 3
Long, G. 118
Lowe, R. 64
Lycurgus 102
MacHale,J. 37
Mallow 58,62
ManseLH.L. 74,75,78,87,93,100
Martineau, J. 118
Maynard,S. 78
McMillian 77
Melbourne 56,64,66,67
Melville's patent ink 115
Mill,J.S. 107
Mitchell, J. 45
Monthly Notices of the Royal Astrono-
mical Society 4
Mulcahy.J. 58,62
Murphy, R. 58,62
Napier, J. 113
Newman, F.W. 79
Newman, J.H. 33
Newton, I. 3,52,54,56,58
O'Higgins.W. 37
Orestes 37
Oriel College 37
Oscott 29
Oxford 80,85,117
Pacius.J. 85,87
partial differential equations 11, 29, 30,
43,95,98
Pascal, B. 3
Pattison, G.S. 118
Paulus Venetus 85,87
Penny Cyclopaedia 4
Persia 102
Peyrard, F. 81,83
Pfaff,J.F. 106,108
Philosophical Magazine 10, 13, 45, 47,
50,51,68
Philosophical Transactions 2, 7, 12, 13, 15
Pierce, C.S. 73
pleurisy 78,79
Poisson, S.D. 2,44,50
PortMadoc 110,112
Powell, B. 80,82
probabilities 3, 17, 31, 32, 34, 47-9,
51,55,58-63,100
probability 1,2,40,45,47,57,62,73
Puissant, L. 70
quadratic form 41
quaternions 10, 66
Queen's College, Oxford 29
Queen's College , Cork 2, 5 , 1 7 , 35 , 5 6
Queen's Colleges (Belfast, Cork, and Galway)
33,91
Quetelet, M.A. 45
Quilp.D. 51,53
Ranyard,A.C 118
Renan, E. 94,101
Rhees, R. 3,32
Robinson Crusoe 94, 102
Robinson, H.C. 118
Rosen, F.A. 118
Rouse Ball, W.W. 4
Royal Medal 2,52,53
Royal Society 7, 11, 13, 14, 15, 32, 52,
56,117,127
Royal Society of Edinburgh 73
Ryall.J. 3,38,56
Scotland 107,110
Sebastopol 65
Shakespeare, W. 62
Shanghai 81
Shelley, P.B. 91
Shylock 86
Smyth, W.H. 118
Society for the Diffusion of Useful Knowl-
edge 3
Solly, T. 29
spherical trigonometry 69, 70, 7 1 , 1 1 3
spiritualism 110,114
Sterling, J. 80
Stevens 69
syllogism 31, 35
Sylvester, J.J. 41,44
Taylor and Walton 53,54
Taylor, Sir G. 3
Thames 53
theorem on definite integration 18-21,
122
theory of series 12
Thomson, W. 29
three-valued logic 30
Todhunter, I. 3, 79, 106-7, 128
Transactions of the Cambridge Philosophical
Society 13,118
Trinity College, Dublin 33,53, 62
Trinity College, Cambridge 3
triple algebra 8, 15
University College London 3, 7, 29, 94,
118
156 INDEX
Veitch.J. 74,101 Wedgewood, H. 79
Vincent de Lerins 40 Whewell.W. 29,75
Wilson, W.R. 64
Waddington 2 Wordsworth, W. 51
Wagner, G. 94, 101
w2?LW?87 Young, J.R. 33, 36, 79
Walsh, J. 40, 43, 44, 47, 53, 54, 105,
107 Zunz,L. 90